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Georgia Standards of Excellence Curriculum Frameworks
GSE Third Grade
Unit 5: Representing and Comparing Fractions
Mathematics
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 2 of 100
All Rights Reserved
TABLE OF CONTENTS
Overview ..........................................................................................................................................3
Standards for Mathematical Practice ...............................................................................................3
Content Standards ............................................................................................................................4
Big Ideas ......................................................................................................................................... 5
Essential Questions ..........................................................................................................................5
Concepts & Skills to Maintain .........................................................................................................6
Strategies for Teaching and Learning ..............................................................................................7
Selected Terms and Symbols ...........................................................................................................8
Tasks ................................................................................................................................................8
Intervention Table ..........................................................................................................................13
FALS ..............................................................................................................................................14
● Exploring Fractions ............................................................................................................15
● Candy Crush.......................................................................................................................20
● Comparing Fractions ..........................................................................................................25
● Strategies For Comparing Fractions ..................................................................................30
● Cupcake Party ....................................................................................................................35
● Using Fraction Strips to Explore the Number Line ...........................................................42
● I Like to Move It! Move It!! ..............................................................................................47
● Pattern Blocks Revisited-Exploring Fractions Further with Pattern Blocks .....................53
● Party Tray...........................................................................................................................58
● Make a Hexagon Game ......................................................................................................67
● Pizzas Made to Order .........................................................................................................72
● Graphing Fractions.............................................................................................................77
● Inch by Inch .......................................................................................................................81
● Measuring to ½ and ¼ Inch ...............................................................................................85
● Trash Can Basketball .........................................................................................................91
Culminating Task
● The Fraction Story Game ...................................................................................................95
IF YOU HAVE NOT READ THE THIRD GRADE CURRICULUM OVERVIEW IN ITS
ENTIRETY PRIOR TO USE OF THIS UNIT, PLEASE STOP AND CLICK HERE:
https://www.georgiastandards.org/Georgia-Standards/Frameworks/3rd-Math-Grade-Level-
Overview.pdf Return to the use of this unit once you’ve completed reading the Curriculum
Overview. Thank you.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 3 of 100
All Rights Reserved
OVERVIEW
In this unit, students will:
● Develop an understanding of fractions, beginning with unit fractions.
● View fractions in general as being built out of unit fractions, and they use fractions along
with visual fraction models to represent parts of a whole.
● Understand that the size of a fractional part is relative to the size of the whole. For
example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a
larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the
ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided
into 5 equal parts. Students are able to use fractions to represent numbers equal to, less
than, and greater than one.
● Solve problems that involve comparing fractions by using visual fraction models and
strategies based on noticing equal numerators or denominators.
● Recognize that the numerator is the top number (term) of a fraction and that it represents
the number of equal-sized parts of a set or whole; recognize that the denominator is the
bottom number (term) of a fraction and that it represents the total number of equal-sized
parts or the total number of objects of the set
● Explain the concept that the larger the denominator, the smaller the size of the piece
● Compare common fractions with like denominators and tell why one fraction is greater
than, less than, or equal to the other
● Represent halves, thirds, fourths, sixths, and eighths using various fraction models.
STANDARDS FOR MATHEMATICAL PRACTICE (SMP)
This section provides examples of learning experiences for this unit that support the development
of the proficiencies described in the Standards for Mathematical Practice. The statements
provided offer a few examples of connections between the Standards for Mathematical Practice
and the content Standards of this unit. The list is not exhaustive and will hopefully prompt
further reflection and discussion.
Students are expected to:
1. Make sense of problems and persevere in solving them. Students make sense of problems
involving fractions.
2. Reason abstractly and quantitatively. Students demonstrate abstract reasoning by
connecting fraction models of shapes with the written form of fractions.
3. Construct viable arguments and critique the reasoning of others. Students construct and
critique arguments regarding fractions by creating or drawing fractional models to prove
answers.
4. Model with mathematics. Students use fraction strips to find equivalent fractions.
5. Use appropriate tools strategically. Students use tiles and drawings to solve the value of a
fraction of a set.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 4 of 100
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6. Attend to precision. Students use vocabulary such as numerator, denominator, and fractions
with increasing precision to discuss their reasoning.
7. Look for and make use of structure. Students compare unit fraction models with various
denominators to reason that as the denominator increases, the size of the unit fraction decreases.
8. Look for and express regularity in repeated reasoning. Students will manipulate tiles to
find the value of a fraction of a set. This will lead to the relationship between fractions and
division.
***Mathematical Practices 1 and 6 should be evident in EVERY lesson***
CONTENT STANDARDS
Develop understanding of fractions as numbers
MGSE3.NF.1 Understand a fraction 1
𝑏 as the quantity formed by 1 part when a whole is
partitioned into b equal parts (unit fraction); understand a fraction 𝑎
𝑏 as the quantity formed by a
parts of size 1
𝑏. For example, 3
4 means there are three
1
4 parts, so 3
4 = 1
4 + 1
4 + 1
4 .
MGSE3.NF.2 Understand a fraction as a number on the number line; represent fractions on a
number line diagram.
a. Represent a fraction 1
𝑏 on a number line diagram by defining the interval from 0 to 1 as
the whole and partitioning it into b equal parts. Recognize that each part has size 1
𝑏.
Recognize that a unit fraction 1
𝑏 is located 1
𝑏 whole unit from 0 on the number line.
b. Represent a non-unit fraction 𝑎
𝑏 on a number line diagram by marking off a lengths of 1
𝑏
(unit fractions) from 0. Recognize that the resulting interval has size 𝑎
𝑏 and that its
endpoint locates the non-unit fraction 𝑎
𝑏 on the number line.
MGSE3.NF.3 Explain equivalence of fractions through reasoning with visual fraction models.
Compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point
on a number line.
b. Recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6, and
8, e.g., 1
2 =
2
4,
4
6 =
2
3. Explain why the fractions are equivalent, e.g., by using a visual
fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole
numbers. Examples: Express 3 in the form 3 = 6
2 (3 wholes is equal to six halves);
recognize that 3
1 = 3; locate 4
4 and 1 at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning
about their size. Recognize that comparisons are valid only when the two fractions refer
to the same whole. Record the results of comparisons with the symbols >, =, or <, and
justify the conclusions, e.g., by using a visual fraction model.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 5 of 100
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MGSE3.MD.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with
several categories. Solve one- and two-step “how many more” and “how many less” problems
using information presented in scaled bar graphs. For example, draw a bar graph in which each
square in the bar graph might represent 5 pets.
MGSE3.MD.4 Generate measurement data by measuring lengths using rulers marked with
halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is
marked off in appropriate units— whole numbers, halves, or quarters.
For more detailed information about unpacking the content standards, unpacking a task, math
routines and rituals, maintenance activities and more, please refer to the Grade Level Overview.
BIG IDEAS
In first grade and second grades, students discuss partitioning and equal shares. Students will
have partitioned circles and rectangles into two, three, and four equal shares. This is the first time
students are understanding/representing fractions through the use of a number line, and
developing deep understanding of fractional parts, sizes, and relationships between fractions.
This is a foundational building block of fractions, which will be extended in future grades.
Students should have ample experiences using the words, halves, thirds, fourths, and quarters,
and the phrases half of, third of, fourth of, and quarter of. Students should also work with the
idea of the whole, which is composed of two halves, four fourths or four quarters, etc.
Example:
How can you and a friend share equally (partition) this piece of paper so that you both have the
same amount of paper to paint a picture?
● Fractional parts are equal shares of a whole or a whole set.
● The more equal sized pieces that form a whole, the smaller the pieces of the whole
become.
● When the numerator and denominator are the same number, the fraction equals one
whole.
● When the wholes are the same size, the smaller the denominator, the larger the pieces.
● The fraction name (half, third, etc) indicates the number of equal parts in the whole.
ESSENTIAL QUESTIONS
● How are fractions used in problem-solving situations?
● How can I compare fractions?
● What are the important features of a unit fraction?
● What relationships can I discover about fractions?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 6 of 100
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CONCEPTS/SKILLS TO MAINTAIN
Third-grade students will have prior knowledge/experience related to the concepts and skills
identified in this unit.
● In first grade, students are expected to partition circles and rectangles into two or four
equal shares, and use the words, halves, half of, a fourth of, and quarter of.
● In second grade, students are expected to partition circles and rectangles into two,
three, or four equal shares, and use the words, halves, thirds, half of, a third of, fourth
of, quarter of.
● Students should also understand that decomposing into more equal shares equals
smaller shares, and that equal shares of identical wholes need not have the same
shape.
Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately,
efficiently, and appropriately. Fluent problem solving does not necessarily mean solving
problems within a certain time limit, though there are reasonable limits on how long computation
should take. Fluency is based on a deep understanding of quantity and number.
Deep Understanding: Teachers teach more than simply “how to get the answer” and instead
support students’ ability to access concepts from a number of perspectives. Therefore, students
are able to see math as more than a set of mnemonics or discrete procedures. Students
demonstrate deep conceptual understanding of foundational mathematics concepts by applying
them to new situations, as well as writing and speaking about their understanding.
Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is
often confused with fluency. Fluency implies a much richer kind of mathematical knowledge and
experience.
Number Sense: Students consider the context of a problem, look at the numbers in a problem,
make a decision about which strategy would be most efficient in each particular problem.
Number sense is not a deep understanding of a single strategy, but rather the ability to think
flexibly between a variety of strategies in context.
Fluent students:
• flexibly use a combination of deep understanding, number sense, and memorization.
• are fluent in the necessary baseline functions in mathematics so that they are able to
spend their thinking and processing time unpacking problems and making meaning from
them.
• are able to articulate their reasoning.
• find solutions through a number of different paths.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 7 of 100
All Rights Reserved
For more about fluency, see: http://www.youcubed.org/wp-
content/uploads/2015/03/FluencyWithoutFear-2015.pdf and:
https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/nctm-timed-
tests.pdf
STRATEGIES FOR TEACHING AND LEARNING
Students need many opportunities to discuss fractional parts using concrete models to develop
familiarity and understanding of fractions. Expectations in this domain are limited to fractions
with denominators 2, 3, 4, 6 and 8.
Understanding that a fraction is a quantity formed by part of a whole is essential to number sense
with fractions. Fractional parts are the building blocks for all fraction concepts. Students need to
relate dividing a shape into equal parts and representing this relationship on a number line, where
the equal parts are between two whole numbers. Help students plot fractions on a number line,
by using the meaning of the fraction. For example, to plot 4/5 on a number line, there are 5 equal
parts with 4 copies of one of the 5 equal parts.
As students counted with whole numbers, they should also count with fractions. Counting equal-
sized parts helps students determine the number of parts it takes to make a whole and recognize
fractions that are equivalent to whole numbers.
Students need to know how big a particular fraction is and can easily recognize which of two
fractions is larger. The fractions must refer to parts of the same whole. Benchmarks such as 1/2
and 1 are also useful in comparing fractions.
Equivalent fractions can be recognized and generated using fraction models. Students should use
different models and decide when to use a particular model. Make transparencies to show how
equivalent fractions measure up on the number line.
Venn diagrams are useful in helping students organize and compare fractions to determine the
relative size of the fractions, such as more than 1/2, exactly 1/2 or less than 1/2. Fraction bars
showing the same sized whole can also be used as models to compare fractions. Students are to
write the results of the comparisons with the symbols >, =, or <, and justify the conclusions with
a model.
For additional assistance with this unit, please watch the unit webinar at:
https://www.georgiastandards.org/Archives/Pages/default.aspx
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 8 of 100
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SELECTED TERMS AND SYMBOLS
The following terms and symbols are often misunderstood. These concepts are not an inclusive
list and should not be taught in isolation. However, due to evidence of frequent difficulty and
misunderstanding associated with these concepts, instructors should pay particular attention to
them and how their students are able to explain and apply them.
The terms below are for teacher reference only and are not to be memorized by the students.
Teachers should present these concepts to students with models and real life examples. Students
should understand the concepts involved and be able to recognize and/or demonstrate them with
words, models, pictures, or numbers. Mathematics Glossary
● bar graph
● common fraction
● decimal fraction
● denominator
● equivalent fraction
● line plot graph
● numerator
● partition
● picture graph
● term
● unit fraction
● whole number
● set
TASKS
The following tasks represent the level of depth, rigor, and complexity expected of all third grade
students. These tasks or a task of similar depth and rigor should be used to demonstrate evidence
of learning. It is important that all standards of a task be addressed throughout the learning
process so that students understand what is expected of them. While some tasks are identified as
a performance task, they also may be used for teaching and learning (constructing task).
Scaffolding Task Tasks that build up to the learning task.
Constructing Task Constructing understanding through deep/rich contextualized problem
solving tasks.
Practice Task Tasks that provide students opportunities to practice skills and concepts.
Performance Task Tasks which may be a formative or summative assessment that checks
for student understanding/misunderstanding and or progress toward the
standard/learning goals at different points during a unit of instruction.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 9 of 100
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Culminating Task Designed to require students to use several concepts learned during the
unit to answer a new or unique situation. Allows students to give
evidence of their own understanding toward the mastery of the standard
and requires them to extend their chain of mathematical reasoning.
Intervention Table The Intervention Table provides links to interventions specific to this
unit. The interventions support students and teachers in filling
foundational gaps revealed as students work through the unit. All
listed interventions are from New Zealand’s Numeracy Project.
Formative
Assessment Lesson
(FAL)
Lessons that support teachers in formative assessment which both reveal
and develop students’ understanding of key mathematical ideas and
applications. These lessons enable teachers and students to monitor in
more detail their progress towards the targets of the standards.
CTE Classroom
Tasks
Designed to demonstrate how the Georgia Standards of Excellence and
Career and Technical Education knowledge and skills can be integrated.
The tasks provide teachers with realistic applications that combine
mathematics and CTE content.
3-Act Task A Three-Act Task is a whole-group mathematics task consisting of 3
distinct parts: an engaging and perplexing Act One, an information and
solution seeking Act Two, and a solution discussion and solution
revealing Act Three. More information along with guidelines for 3-Act
Tasks may be found in the Guide to Three-Act Tasks on
georgiastandards.org and the K-5 Georgia Mathematics Wiki.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 10 of 100
All Rights Reserved
Task Name Task Type
Grouping Strategy Content Addressed Standard(s)
Description
Exploring Fractions
Scaffolding Task
Individual/Small Group
Task
Naming fractional pieces,
length model MGSE3.NF.1
In this lesson students will use strips of paper
to create fraction bar models that they can
refer back to and utilize throughout the unit
Candy Crush!
Constructing Task
Partner/Small Group
Task
Naming the value of a
fraction of a set MGSE3.NF.1
In this lesson students will find the value of
fractions of sets.
Comparing Fractions
Scaffolding Task
Individual/Small Group
Task
Naming fractional pieces,
exploring inequalities,
length model
MGSE3.NF.3
Students will create models of fractions that
they can manipulate to find equivalent
fractions.
Strategies for
Comparing Fractions
Scaffolding Task
Individual/Small Group
Task
Naming fractional pieces,
exploring inequalities,
length model
MGSE3.NF.3
Students will use their fraction bars from the
previous lesson to find inequalities and
express those inequalities as number
sentences.
Cupcake Party 3-Act Task
Whole Group
Divide and label fractional
parts
MGSE3.NF.1
MGSE3.NF.3
In this task, students will watch a Vimeo and tell
what they noticed. Next, they will be asked to
discuss what they wonder about or are curious
about. Students will then use mathematics to
answer their own questions.
Using Fraction Strips
to Explore the Number
Line
Constructing Task
Individual/Small Group
Task
Create a number line,
explore fractions between 0
– 1, length model
MGSE3.NF.2
Students create fraction number lines using
strips of paper and use the number lines to
find equalities and inequalities.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 11 of 100
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Representing Fractions
on a Number Line
Formative Assessment
Lesson
Representing fractions on a
number line
MGSE3.NF.1
MGSE3.NF.2
This formative assessment is designed to be
implemented approximately two-thirds of the
way through the instructional unit.
I Like to Move It!
Move It!!
Constructing Task
Partner/Small Group
Task
Placing fractions on number
lines, iterating unit fractions
MGSE3.NF.1
MGSE3.NF.2
MGSE3.OA.3
Students will count unit fraction on number
lines.
Pattern Block Fractions
Revisited
Constructing Task
Partner/Small Group
Task
Apply skills in identifying
fractional parts of a whole MGSE3.NF.1
Students will partition pattern blocks using
various sized wholes.
Party Tray 3-Act Task
Whole Group
Apply skills in identifying
fractional parts of a whole
Practice using fractional
parts to make a whole
MGSE3.NF.1
In this task, students will look at a picture and tell
what they noticed. Next, they will be asked to
discuss what they wonder about or are curious
about. Students will then use mathematics to
answer their own questions.
Make a Hexagon Game
Practicing Task
Partner/Small Group
Task
Practice using fractional
parts to make a whole MGSE3.NF.1
Students will play a game where they create a
fraction with dice and build their fraction on
hexagons using pattern blocks.
Pizzas Made to Order Practicing Task
Individual Task
Divide and label fractional
parts
MGSE3.NF.1
MGSE3.NF.3
Students will fill pizza orders by representing
the ordered ingredients on the appropriate
fractional parts of a pizza cut-out.
Graphing Fractions
Constructing Task
Individual/Small Group
Task
Representing data as
fractions, graphing data
MGSE3.MD.3
MGSE3.NF.1
Students will create graphs and then identify
the fractional representation of each bar of
data and create questions that could be
answered using the data in their graphs
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 12 of 100
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Inch by Inch
Constructing Task
Whole/Small Group
Task
Using nonstandard units of
measurement to measure to
the ¼ and ½ inch.
MGSE3.MD.4
Students will create ruler using strips of paper
and then create a line plot graph to collect and
record the data of the objects they have
measured to the nearest ¼ inch throughout the
class.
Measuring to the ½ and
¼ Inch
Constructing Task
Individual/Small Group
Task
Using a number line to
measure and organize data
MGSE3.MD.4
MGSE3.NF.2
MGSE3.NF.3
Students will measure objects to the nearest ½
and ¼ inch. Students will order their
measurements from shortest to longest and
will create a line-plot graph to represent their
data.
Trash Can Basketball
Practice Task
Partner/Small Group
Task
Representing data as
fractions
MGSE3.NF.1
MGSE3.NF.2
MGSE3.NF.3
Students will play a game where they write a
fraction that represents the number of shots
made and then create a poster that represents
their results using an inequality.
The Fraction Story
Game
Culminating Task
Individual/Small Group
Task
Create a fraction game
using story problems
MGSE3.NF.1
MGSE3.NF.2
MGSE3.NF.3
Students create a game while reviewing
all the different aspects of fractions they
have studied.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 13 of 100
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INTERVENTION TABLE
The Intervention Table provides links to interventions specific to this unit. The interventions
support students and teachers in filling foundational gaps revealed as students work through the
unit. All listed interventions are from New Zealand’s Numeracy Project
Cluster of Standards Name of
Intervention
Snapshot of summary or
Student I can statement. . .
Materials
Master
Numbers and
Operations –
Fractions
Develop
understanding
of fractions as
numbers
MGSE3.NF.1
MGSE3.NF.2
MGSE3.NF.3
Unit
Fractions Compare and order unit fractions
Creating
Fractions
Identify the symbols for halves,
quarters, thirds, fifths, and tenths
including fractions greater than 1. MM 4-20
More Geo
Board
Fractions
Use geoboards to create fractions of
regions MM 4-20
Fractions in a
Whole
In this task, students determine how
many unit fractions are in a whole
Fraction
Pieces
Identify the symbols for halves,
quarters, thirds, fifths, and tenths
including fractions greater than 1
MM 4-19
Who Has
More Cake Order and compare unit fractions MM 4-19
Fraction Bits
and Parts
A series of lessons that assist
students in making, naming, and
recognizing fractions
CM 1
CM 2
CM 3
CM 4
CM 5
CM 6
CM 7
CM 8
Playdough
0-1 on the
number line Place fractions on a number line
Trains Place unit fractions larger than one
on a number line
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 14 of 100
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FORMATIVE ASSESSMENT LESSONS (FALS)
Formative Assessment Lessons are designed for teachers to use in order to target specific
strengths and weaknesses in their students’ mathematical thinking in different areas. A
Formative Assessment Lesson (FAL) includes a short task that is designed to target
mathematical areas specific to a range of tasks from the unit. Teachers should give the task in
advance of the delineated tasks and the teacher should use the information from the assessment
task to differentiate the material to fit the needs of the students. The initial task should not be
graded. It is to be used to guide instruction.
Teachers are to use the following Formative Assessment Lessons (FALS) Chart to help them
determine the areas of strengths and weaknesses of their students in particular areas within the
unit. The chart lists each FAL to use for a specific task or task along with the content addresses.
Formative Assessments Content Addressed Pacing
(Use before and after this task)
ELEMENTARY
FORMATIVE
ASSESSMENT LESSONS
Representing Fractions on
a Number Line
I Like to Move It! Move It!!
Although the units in this instructional framework emphasize key standards and big ideas at
specific times of the year, routine topics such as estimation, mental computation, and basic
computation facts should be addressed on an ongoing basis. Ideas related to the eight practice
standards should be addressed constantly as well. This unit provides much needed content
information and excellent learning activities. However, the intent of the framework is not to
provide a comprehensive resource for the implementation of all standards in Unit 5. A variety of
resources should be utilized to supplement this unit. The tasks in this unit framework illustrate
the types of learning activities that should be utilized from a variety of sources. To assure that
this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the
“Strategies for Teaching and Learning” and the tasks listed under “Big Ideas” be reviewed
early in the planning process.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 15 of 100
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SCAFFOLDING TASK: EXPLORING FRACTIONS Return to Task Table Adapted from NCTM Illuminations
APPROXIMATE TIME: 2 class periods
In this lesson students will use strips of paper to create fraction bar models that they can
refer back to and utilize throughout the unit.
CONTENT STANDARDS
MGSE3.NF.1 Understand a fraction 1
𝑏 as the quantity formed by 1 part when a whole is
partitioned into b equal parts (unit fraction); understand a fraction 𝑎
𝑏 as the quantity formed by a
parts of size 1
𝑏. For example, 3
4 means there are three
1
4 parts, so 3
4 = 1
4 + 1
4 + 1
4 .
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Before the activity, be sure the children understand the concept of equal parts. Use pieces of
different shaped paper (piece of construction paper, coffee filter, 8 ½ inch square cut from a
piece of copy paper, 1/2 sheet of copy paper cut vertically, etc.) to demonstrate folding into
equal-sized pieces. For some of the students to understand “equal-sized” you may have to
cut and match the pieces, demonstrating that they are the same size. The use of different
models, such as fraction bars and number lines, allows students to compare unit fractions
and to reason about their sizes.
COMMON MISCONCEPTIONS
Students do not understand that when partitioning a whole shape, number line, or a set into unit
fractions, the intervals must be equal. Students think all shapes can be divided the same way. To
fix this misconception, students should have plenty of experiences with partitioning varying
shapes and sets of shapes into equal parts.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 16 of 100
All Rights Reserved
ESSENTIAL QUESTIONS
● What is a fraction?
● How can I represent fractions of different sizes?
● What relationships can I discover about fractions?
MATERIALS
● Exploring Fractions task sheet
● 9” x 12” sheets of paper in six different colors (cut into 1” x 12” strips) Each child will
need 6 strips, one of each color.
● Scissors
● File folder (1 for each child)
● Glue or tape
GROUPING
Individual/Partner Task
NUMBER TALK
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Part I (SMP 1, 3, 4, 5, 6, 7)
To assess prior knowledge, ask students to create a list of ways they use fractions in their daily
lives. Some examples may include dividing a snack in half (1/2), eating 3/8 of a pizza, using
measuring cups or spoons while baking, money (half a dollar), time (quarter of an hour).
Read aloud and discuss, Whole-y Cow! by Taryn Souders (or another book about the concept of
fractions).
To begin the lesson, give students six strips of paper in six different colors. Specify one color
and have students hold up one strip of this color. Tell students that this strip will represent the
whole. Have students write “one whole” on the fraction strip. The term whole is included in the
labeling instead of 1 because it helps eliminate confusion between the numeral 1 in fractions
such as ½.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 17 of 100
All Rights Reserved
Next, ask students to pick a second strip and fold it into two equal pieces. Have students draw a
line on the fold. Ask students what they think each of these strips should be called (one-half or
½). It is important, here, for students to understand how fractions are named. Discuss the names
numerator and denominator with students. Have students label their strips accordingly using both
the word and the fractional representation. Label both sides of the strip “1/2 one-half.”
Have students take out another strip, fold it in half twice, and divide it into four congruent pieces.
Ask them what they think each of these strips should be called (one-fourth or ¼). Students
should draw lines on the folds and label the strips using both the word and the fraction. Label all
four sections of the strip “1/4 one-fourth”. Repeat the process of folding in half and naming
eighths.
Students will take out another strip, fold it in thirds and divide it into three congruent pieces.
Ask them what they think each of these strips should be called (one-third or 1/3). Have students
draw lines on the folds and label the strips using both the word and the fraction. Label all three
sections of the strip “1/3 one-third”. Repeat the process of folding in thirds and then in half to
create sixths. Label each section “1/6 one-sixth.”
After folding and labeling strips of paper for the whole, halves, thirds, fourths, sixths, and
eighths, ask students to glue or tape the strips on their file folder in order (largest fractional
pieces to smallest fractional pieces). Make sure the students line up the strips evenly so that they
begin to see equivalences. Suggestion: Secure the ½ strip first with the half mark on the crease
in the file folder. Place every other paper strip in line with one-half.
Part II (SMP 7)
Arrange students in small groups of 2-3 students. Give them approximately ten minutes to write
down their observations about the fraction strips. Have each group share some of their
comments. Lead the groups to consider questions such as:
● How many halves does it take to make a whole strip?
● How many thirds does it take to equal one whole?
● How many fourths, sixths, eighths?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 18 of 100
All Rights Reserved
Part III (SMP 1, 2, 3, 4, 5, 7, 8)
Have students work in small groups to answer the questions below. The teacher should monitor
the groups, asking questions, and encouraging students to explore the concept of fractions. Have
groups (at least 2-3) share their solution to question number seven. Try to pick groups who
presented different ways of dividing the sandwich.
FORMATIVE ASSESSMENT QUESTIONS
● Is your strip folded into equal parts? How do you know?
● What relationships did you discover about fractions?
● What does the numerator represent?
● What does the denominator represent?
DIFFERENTIATION
Extension
● Have students create additional fraction strips and write about relationships.
Intervention
● Use ready-made Fraction Tiles or Virtual Manipulatives.
● Use partitioning of shapes to link to this task. Show how the student is partitioning
rectangles as they fold the strips.
● Class Fractions
Use a group of students as the whole – for example, six students if you want to work
on 1/3s, 1/2s, and 1/6s. Ask students, “What fraction of our friends (are wearing
tennis shoes, have brown hair, etc.)?” Change the number of people over time.
• Adapted from Elementary and Middle School Mathematics: Teaching
Developmentally by John A. Van de Walle, Karen S. Karp, and Jennifer M. Bay-
Williams, p. 290.
Intervention Table
TECHNOLOGY RESOURCES
• http://www.kidsnumbers.com/turkey-terminator-math-game.php
http://www.visualfractions.com/
https://www.conceptuamath.com/app/tool-library Conceptua Learning Tools (Fraction
Tab) are great for both parents and teachers while working on fraction concepts.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 19 of 100
All Rights Reserved
Name: ___________________________ Date: _______________
EXPLORING FRACTIONS
(Adapted from a lesson by Angela Lacey Hester, Floyd County, GA)
1. Using complete sentences and math words, write 3 observations you and your group
made about the Fraction Strips.
Use your Fraction Strips to answer the following questions.
2. How many thirds does it take to equal one whole?
3. How many sixths does it take to equal one whole?
4. What do you think three 1/8 strips might be called? How would you write that
fraction?
5. If you made a 1/9 fraction strip, how many ninths would it take to make a whole?
Put on your thinking caps…
6. What would a 1/10 Fraction Strip look like? Sketch and label the Fraction Strip in the
space below.
7. Pretend you are having a party for 6 people. For refreshments, you are serving a 12”
sub sandwich. On the back of this paper, draw and label a 12” sub (just like your Fraction
Strips). Show how you would equally divide the sandwich for 6 people. Use pictures,
words, and numbers to explain your reasoning.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 20 of 100
All Rights Reserved
CONSTRUCTING TASK: CANDY CRUSH! Return to Task Table
APPROXIMATE TIME: 1 Class period
In this lesson students will find the value of fractions of sets.
CONTENT STANDARDS
MGSE3.NF.1 Understand a fraction 1
𝑏 as the quantity formed by 1 part when a whole is
partitioned into b equal parts (unit fraction); understand a fraction 𝑎
𝑏 as the quantity formed by a
parts of size 1
𝑏. For example, 3
4 means there are three
1
4 parts, so 3
4 = 1
4 + 1
4 + 1
4 .
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
BACKGROUND KNOWLEDGE
Students are introduced to fractions in 1st and 2nd grade by partitioning circles and rectangles
into two, three, or four equal shares, describing the shares using the words halves, thirds, fourths,
and quarters. This task will allow students to partition sets into fractional parts and will provide
a foundation for exploring equivalent fractions in the next lessons of this unit. This task also
links fractions with the concept of division which they have learned about in an earlier 3rd grade
unit.
Understanding that parts of a whole must be partitioned into equal-sized shares across different
models (shapes, number lines, and sets) is an important step in conceptualizing fractions and
provides a foundation for exploring equivalence tasks, which are prerequisite to performing
fraction operations (Cramer & Whitney, 2010)
COMMON MISCONCEPTIONS
Students plot points based on understanding fractions as whole numbers instead of fractional
parts. For example: Students order fractions using the numerator or students order unit fractions
by the denominator. Students also see the numbers in fractions as two unrelated whole numbers
separated by a line.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 21 of 100
All Rights Reserved
ESSENTIAL QUESTIONS
● What represents the denominator in a set?
● What represents the numerator in a set?
MATERIALS
● counters
GROUPING
Partner/Small Group
NUMBER TALKS
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION (SMP 1, 2, 3, 4, 5, 6, 7)
Begin by asking students to think about and discuss the meaning of the numerator and
denominator in the fraction ½. It is essential that students have a clear understanding of the
meaning of the digits in a fraction to complete this task. It may also be helpful to display the
class-created definitions of numerator and denominator to refer to during the lesson.
In this task, students are rewarded for their good behavior! They get to choose the kind of candy
they want based on their fractional calculations. To help guide students’ thinking as needed,
refer them back to the meaning of the denominator and the numerator. Students will then answer
the task questions.
FORMATIVE ASSESSMENT QUESTIONS
● What does the denominator represent in this problem?
● What does the numerator represent in this problem?
● Do fractions always represent the same amount? Why or why not?
DIFFERENTIATION
Extension
● Students can find other fractional parts of the available candy. If they are comfortable
finding unit fractions, have them find fractions other than unit fractions. Examples: ⅔, ⅗
etc. Have these students explore what happens when the number in the set is not a
multiple of the denominator. For example, what happens when you have 24 pieces of
candy that you want to divide into fifths?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 22 of 100
All Rights Reserved
Intervention
● Provide struggling students with a sheet of paper that is divided into equal groups that is
represented by the denominator in their problem. Have them count out the number of
counters in the set in their problem and divide them evenly into the groups.
● Intervention Table
TECHNOLOGY RESOURCES
https://www.conceptuamath.com/app/tool-library Conceptua Learning Tools (Fraction Tab) are
great for both parents and teachers while working on fraction concepts
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 23 of 100
All Rights Reserved
Name: ____________________________ Date: ____________________
Candy Crush!
Directions: You are being rewarded for your good classroom behavior! Use counters
to help you solve and then draw a picture to justify each answer.
Jar #1 contains 24 pieces of Twizzlers. How many Twizzlers will you get if you can
have ¼ of them?
Jar #2 contains 12 Hershey’s Kisses. How many Hershey’s Kisses can you get if you
can have ½ of them?
Jar #3 contains 21 Gummie Bears. How many Gummie Bears can you get if you can
have ⅓ of them?
Jar #4 contains 16 Skittles. How many Skittles can you get if you can have ½ of
them?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 24 of 100
All Rights Reserved
Name: __________________________ Date: _____________________
Candy Crush! - Questions
1. Write a number sentence that represents how you solved each problem.
Jar #1 ________________________
Jar #2________________________
Jar #3________________________
Jar #4 ________________________
What operation is related to fractions? _________________
2. Does ½ always represent the same value? Explain your thinking.
3. Which candy will you choose? Explain your thinking.
4. Stanley chose 5 pieces of peppermints. If 5 pieces represents ¼ of all the
peppermints, how many peppermints were there altogether?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 25 of 100
All Rights Reserved
SCAFFOLDING TASK: COMPARING FRACTIONS Return to Task Table
Adapted from NCTM Illuminations
APPROXIMATE TIME: 2 class periods
Students will create models of fractions that they can manipulate to find
equivalent fractions.
CONTENT STANDARDS
MGSE3.NF.3 Explain equivalence of fractions through reasoning with visual fraction models.
Compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point
on a number line.
b. Recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6, and
8, e.g., 1
2 =
2
4,
4
6 =
2
3. Explain why the fractions are equivalent, e.g., by using a visual
fraction model.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Counting fractional parts is the groundwork for comparing and understanding the two parts of
fractions. When developing this thinking, it is useful to display fraction pie pieces and count
them together as a class. For example, using the fractions 1/4, 2/4, 3/4, 4/4, and 5/4, the class
can discuss the relationship the fractions have with one whole. (Van de Walle, p. 138)
COMMON MISCONCEPTIONS
Students do not understand the importance of the whole of a fraction and identifying it. For
example, students may use a fixed size of ¼ based on the manipulatives used or previous
experience with a ruler.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 26 of 100
All Rights Reserved
ESSENTIAL QUESTIONS
● What relationships can I discover about fractions?
● How can I compare fractions?
● What equivalent groups of fractions can I discover using Fraction
Strips?
MATERIALS
● Comparing Fractions task sheet
● 9” x 12” sheets of paper in six different colors (cut into 1” x 12” strips) Each child will
need 6 strips, one of each color.
● Scissors
GROUPING
Partner/Small Group
NUMBER TALKS
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Part I (SMP 1, 4, 5, 6, 7)
Give students six strips of paper in six different colors. Repeat the Fraction Strip folding
and labeling activity from the Exploring Fractions Task. This time, ask students to separate the
Fraction Strips by cutting on the folds giving them 2 - ½ strips, 3 – 1/3 strips, and so forth. Give
each student a plastic sandwich bag or envelope to store the strips. (You can also use fraction
bars)
Arrange students in small groups of 2-3 students. Give them approximately ten minutes to write
down their observations about the separated Fraction Strips. Have each group share some of
their comments. Lead the groups to consider questions such as:
● Do you see any special relationships among the different colored strips?
● Place a ½ strip on your desk. How many strips or combinations of strips are the same
size as ½?
● When fractions are the same size, they are called equivalent. What other equivalent sets
of fractions can you create?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 27 of 100
All Rights Reserved
Have students line up their fraction strips and find as many relationships as they can. For
instance, they might notice that three of the 1/6 pieces are equal to four of the 1/8 pieces, or that
two of the 1/3 pieces are equal to four of the 1/6 pieces. Have students record these relationships
on paper. When they have finished, have them share the relationships they have discovered.
Record the relationships on chart paper and discuss.
Students will notice that one whole is the same as 2/2, 4/4, 8/8, 3/3, or 6/6. Another example
includes the relationship between ½, 2/4, 4/8, and 3/6. Tell students that when fraction strips are
the same length, they represent equivalent fractions. Students may also notice that for each of
these fractions, the numerator is ½ of the denominator.
Part II (SMP 1, 2, 4, 5, 7, 8)
Students will work in small groups to answer the questions in the activity sheet. The teacher
should monitor the groups, asking questions, and encouraging students to explore the concept of
fractions.
Have groups (at least 2-3) share their solution to question numbers 6 and 7. Try to pick groups
who presented different ways of solving the problems. After this lesson, have students store
their Fraction Strips in a plastic sandwich bag.
Part III (SMP 1, 3, 4, 5, 6, 7)
Students can practice comparing fractions using the following activity adapted from Elementary
and Middle School Mathematics: Teaching Developmentally by John A. Van de Walle, Karen
S. Karp, and Jennifer M. Bay-Williams, p. 290.
The friends below are playing red light-green light. Who is winning? Use your
fraction strips to determine how far each friend has moved.
Mary – ¾ Harry – ½ Larry – 5/6
Sam – 5/8 Michael – 5/9 Angie – 2/3
FORMATIVE ASSESSMENT QUESTIONS
● What relationships did you discover about fractions?
● How can you compare fractions?
● What equivalent groups of fractions did you discover?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 28 of 100
All Rights Reserved
DIFFERENTIATION
Extension
● Students can use coffee filters, paper plates, or other objects to create
different models to illustrate inequalities.
Intervention
● Use ready-made Fraction Tiles or Virtual Manipulatives.
● Intervention Table
TECHNOLOGY RESOURCES
• http://www.mathplayground.com/Scale_Fractions.html
• http://illuminations.nctm.org/ActivityDetail.aspx?ID=80
• https://www.conceptuamath.com/app/tool-library Conceptua Learning Tools (Fraction
Tab) are great for both parents and teachers while working on fraction concepts.
• http://www.gregtangmath.com/satisfraction This game allows the user to filter by various
comparison strategies (e.g., common numerator, common denominator, etc.) and requires
students to vary between picking the largest and smallest fractions.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 29 of 100
All Rights Reserved
Name: _______________________________ Date: _______________
COMPARING FRACTIONS
(Adapted from a Learning Task by Angela Lacey Hester, Floyd County, GA)
1. Using complete sentences and math words, write 3 observations you and your group made about
the Fraction Strips.
Use your Fraction Strips to answer the following questions.
2. What fraction is equivalent to 2 of your 1/4 strips?
3. What fraction is equivalent to 3/6?
4. What fraction is equivalent to 6/8?
5. If you had made a fraction strip for 1/10s, how many tenths would it take to make to equal 1/2?
Put on your thinking caps….
6. In the space below, draw a Fraction Strip divided into fourths. Draw 2 additional shapes divided
into fourths. Make one of your drawings a real-life example of something you might partition
(divide) into fourths.
7. Pretend it is 7:30 a.m. Math Class begins at 8:00 a.m. Ashley says class starts in 30 minutes.
Harrison says class starts in half an hour. Which child is correct? On the back of this page, draw a
picture and write 2-3 sentences to explain your answer.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 30 of 100
All Rights Reserved
SCAFFOLDING TASK: STRATEGIES FOR COMPARING FRACTIONS
Return to Task Table Adapted from NCTM Illuminations
APPROXIMATE TIME: 2 class periods
Students will use their fraction bars from the previous lesson to find inequalities and
express those inequalities as number sentences.
CONTENT STANDARDS
MGSE3.NF.3 Explain equivalence of fractions through reasoning with visual fraction models.
Compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point
on a number line.
b. Recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6, and
8, e.g., 1
2 =
2
4,
4
6 =
2
3. Explain why the fractions are equivalent, e.g., by using a visual
fraction model.
d. Compare two fractions with the same numerator or the same denominator by reasoning
about their size. Recognize that comparisons are valid only when the two fractions refer
to the same whole. Record the results of comparisons with the symbols >, =, or <, and
justify the conclusions, e.g., by using a visual fraction model.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Counting fractional parts is the groundwork for comparing and understanding the two parts of
fractions. When developing this thinking, it is useful to display fraction pie pieces and count
them together as a class. For example, using the fractions 1/4, 2/4, 3/4, 4/4, and 5/4, the class
can discuss the relationship the fractions have with one whole. (Van De Walle, p. 138)
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 31 of 100
All Rights Reserved
COMMON MISCONCEPTIONS
Students do not understand the importance of the whole of a fraction and identifying it. For
example, students may use a fixed size of ¼ based on the manipulatives used or previous
experience with a ruler.
ESSENTIAL QUESTIONS
● How can I show that one fraction is greater (or less) than another using my Fraction
Strips?
● How can I compare fractions when they have the same denominators?
● How can I compare fractions when they have the same numerators?
MATERIALS
● Strategies for Comparing Fractions task sheet
● Fraction strips from previous task
GROUPING
Partner/Small Group
NUMBER TALKS
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Students will need their six strips of paper in six different colors from the previous task. Briefly
review concepts covered in previous lessons.
Part I (SMP 1 and 4)
Guide students to compare fraction strips, this time encourage students to compare individual
strips and explore which ones are longer and shorter. Arrange students in small groups of 2-3
students. Give them approximately ten minutes to write down their observations from
comparing the Fraction Strips. Have each group share some of their comments. Lead the groups
to consider questions such as:
● What special relationships do you notice among the different colored strips?
● Place a ½ strip on your desk. How many strips are less than ½?
● Place a 1/8 strip on your desk. How many strips are less than 1/8?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 32 of 100
All Rights Reserved
Part II (SMP 1, 4, 5, 7)
Instruct students to compare two fraction strips: ½ and ¼. Discuss which one is longer and
which one is shorter. Have students discuss how they might write the inequality statements: ½ >
¼ and ¼ < ½. Guide them to the use of the symbols if they don’t do this independently. Repeat
the activity with several additional fraction strips. Be sure to include equivalent fractions such as
½ = 2/4.
Part III (SMP 1, 2, 3, 4, 5, 6, 7, 8)
Same Denominators/Different Numerator:
Have students work in groups of 4. Ask them to arrange 3 groups of fractions in their work
space. In row one, place 1 - 1/3 strip. In row two, place 2 – 1/3 strips. In row three, place 3 –
1/3 strips. On a sheet of paper, have the students write the names of the strips in order from
shortest to longest (1/3, 2/3, 3/3). Encourage students to look for patterns. What do they observe
about the denominators? (All are three.) What do they observe about the denominators? (They
go in order getting larger each time.) How do the numerators relate to the size of the fraction
strips? (The larger the numerator, the larger the strip of paper.) Why? (The larger the
numerator, the more equal sized pieces you have.)
Ask students to repeat the above activity with their 1/4th strips. Discuss the students’
observations.
Same Numerator/Different Denominator:
Have students place one of each color Fraction Strip in their work space. At this time, do not
include one whole. Ask students to arrange the strips from shortest to longest. Have the students
write the names of the strips in order from shortest to longest (1/8, 1/6, ¼, 1/3, 1/2). Encourage
students to look for patterns. What do they observe about the numerators? (All are one.) What
do they observe about the denominators? (They go in order getting smaller each time.) How do
the denominators relate to the size of the fraction strips? (The smaller the denominator, the
larger the strip of paper.) Why? (The larger the denominator, the more pieces it takes to make
the whole.)
Repeat this activity using 2 of each strip. Ask students to once again arrange the pairs of strips in
order from smallest to largest (2/8, 2/6, 2/4, 2/3, 2/2). Discuss the students’ observations.
Part IV (SMP 1, 2, 3, 4, 5, 6, 7, 8)
Have students work in small groups to answer the questions in the task sheet. The teacher should
monitor the groups, asking questions, and encouraging students to explore the concept of
fractions.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 33 of 100
All Rights Reserved
At least two or three groups should share their solution to question number 6. Try to pick groups
who presented different ways of solving the problems. After this lesson, have students store
their Fraction Strips in their sandwich bag.
FORMATIVE ASSESSMENT QUESTIONS
● What relationships did you discover about fractions?
● How can you compare fractions with the same denominators?
● How can you compare fractions with the same numerators?
DIFFERENTIATION
Extension
● Have students write a set of guidelines and illustrations for comparing
fractions and share with a peer.
Intervention
● Use ready-made Fraction Tiles or Virtual Manipulatives.
● Ordering Unit Fractions
List a set of unit fractions such as ½, 1/3, 1/8, 1/5. Ask children to put the
fractions in order from least to greatest. Challenge students to defend the way
they ordered fractions. Ask them to illustrate their idea using fraction strips or
other models.
Repeat the activity using fractions with the same denominators
such as 3/5, 2/5, 5/5, 4/5, 1/5.
Adapted from Elementary and Middle School Mathematics: Teaching Developmentally
By John A. Van de Walle, Karen S. Karp, and Jennifer M. Bay-Williams, p. 300.
• Intervention Table
TECHNOLOGY RESOURCES
http://www.gamequarium.com/fractions.html
http://www.learningplanet.com/sam/ff/index.aspThis site has both teacher and student
activities .
https://www.conceptuamath.com/app/tool-library Conceptua Learning Tools (Fraction Tab)
are great for both parents and teachers while working on fraction concepts
http://www.gregtangmath.com/satisfraction This game allows the user to filter by various
comparison strategies (e.g., common numerator, common denominator, etc.) and requires
students to vary between picking the largest and smallest fractions.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 34 of 100
All Rights Reserved
Name: _______________________________ Date: _______________
STRATEGIES FOR COMPARING FRACTIONS
(Adapted from a Learning Task by Angela Lacey Hester, Floyd County, GA)
1. Using complete sentences and math words, write 3 observations you and your group made about
fraction inequalities, comparing fractions with the same denominators, and comparing fractions with
the same numerators.
Use your Fraction Strips to answer the following questions.
2. Write an inequality statement for the fractions 1
2 and
3
8.
3. Write two inequality statements using 1
6,
1
8,
1
3,
1
2, and
1
4.
Put on your thinking caps….
4. Pretend you had fraction strips for 1
5. Put the following fractions in order from smallest to
largest: 1
5,
5
5,
3
5,
4
5, and
2
5. Draw a picture below to help explain your answer.
5. Using what you have learned about comparing fractions, put the following fractions in order from
least to greatest: 3
4,
3
7 ,
3
3, and
3
8. Draw a picture below to help explain your answer. Stretch your
brain- where would 3
2 go? What might
3
2 look like?
6. For the class party, Robin and Shawn each made a pan of brownies. Their pans were exactly the
same size. Robin sliced her brownies into 9 pieces. Shawn sliced his into 12 pieces. Which student
had the largest brownie pieces? On the back of this paper, make a sketch of Robin and Shawn’s
brownies. Explain your reasoning using words, pictures, and numbers.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 35 of 100
All Rights Reserved
3ACT TASK: CUPCAKE PARTY Return to Task Table
APPROXIMATE TIME: 1 class period
CONTENT STANDARDS
MGSE3.NF.1 Understand a fraction 1
𝑏 as the quantity formed by 1 part when a whole is
partitioned into b equal parts (unit fraction); understand a fraction 𝑎
𝑏 as the quantity formed by a
parts of size 1
𝑏. For example, 3
4 means there are three
1
4 parts, so 3
4 = 1
4 + 1
4 + 1
4 .
MGSE3.NF.3 Explain equivalence of fractions through reasoning with visual fraction models.
Compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point
on a number line.
b. Recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6, and
8, e.g., 1
2 =
2
4,
4
6 =
2
3. Explain why the fractions are equivalent, e.g., by using a visual
fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole
numbers. Examples: Express 3 in the form 3 = 6
2 (3 wholes is equal to six halves);
recognize that 3
1 = 3; locate 4
4 and 1 at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning
about their size. Recognize that comparisons are valid only when the two fractions refer
to the same whole. Record the results of comparisons with the symbols >, =, or <, and
justify the conclusions, e.g., by using a visual fraction model.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them. Students must make sense of
the problem by identifying what information they need to solve their question.
2. Reason abstractly and quantitatively. Students are asked to make an estimate (high
and low).
3. Construct viable arguments and critique the reasoning of others. After writing down
their own questions, students discuss their question with partners, creating the
opportunity to construct the argument of why they chose their question, as well as
critiquing the questions that others came up with.
4. Model with mathematics. Once given the information, the students use that
information to develop a mathematical model to solve their question. Students should
use model to justify their reasoning.
5. Use appropriate tools strategically. Students use number lines, drawings, or
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 36 of 100
All Rights Reserved
manipulatives to help them answer their main questions regarding fractions.
6. Attend to precision. Students use precise language when they discuss their reasoning
with others.
8. Look for and express regularity in repeated reasoning. Students continually evaluate
their work by asking themselves “Does this make sense?”
ESSENTIAL QUESTIONS
In order to maintain a student-inquiry-based approach to this task, it may be beneficial to wait
until Act 2 to share the EQ’s with your students. By doing this, students will be allowed the
opportunity to be very creative with their thinking in Act 1. By sharing the EQ’s in Act 2, you
will be able to narrow the focus of inquiry so that the outcome results in student learning directly
related to the content standards aligned with this task.
● What does it mean to partition a shape into parts?
● What are the important features of a unit fraction?
MATERIALS
● Act 1 video- https://vimeo.com/94999740
● Student recording sheet
GROUPING
Individual/Partner and or Small Group
BACKGROUND KNOWLEDGE:
This task follows the 3-Act Math Task format originally developed by Dan Meyer. More
information on this type of task may be found at http://blog.mrmeyer.com/category/3acts/. A
Three-Act Task is a whole-group mathematics task consisting of 3 distinct parts: an engaging
and perplexing Act One, an information and solution seeking Act Two, and a solution discussion
and solution revealing Act Three. More information along with guidelines for 3-Act Tasks may
be found in the Guide to Three-Act Tasks on georgiastandards.org and the K-5 Georgia
Mathematics Wiki.
Up to this point, students have not worked with written fractions. They have partitioned circles
and rectangles into equal shares using the words halves, thirds, half of, a third of, etc., but have
not written fractions in fractional form ( ½ ).
Concepts about fractions are basic to mathematics but can pose challenges for students. In
elementary schools, the most frequently used fraction models are the region and set models. This
lesson exposes students to the region model and gives an opportunity for them to develop a
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 37 of 100
All Rights Reserved
thorough understanding of this model in multiple applications. As students work with a variety
of fraction models in contexts that promote reasoning and problem solving, they develop a more
thorough understanding of fractions and the relationships among them.
In this task, students may need scaffolds from the teacher. Students who struggle to see how to
split the cupcakes evenly among 4 students will need prompts through questioning. When
partitioning or sharing more than one object, one way to scaffold the activity for students is to
ask, “How can you share 1 ______ (whatever the object is)?” In this particular task, it may be
necessary to ask students how they would share 1 cupcake, and then have them apply that
understanding to the 3 cupcakes. By requiring students to think about how to share 1 object, they
are explicitly drawn to the unit fraction.
COMMON MISCONCEPTIONS:
When partitioning a whole shape into parts, it is important to understand that the size of the
parts must be equal, but the shape of the parts do not have to be the same. This task allows
students to experience fractional parts that are not necessarily the same shape, but are the
same size.
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will view the video and tell what they noticed. Next, they will be asked to
discuss what they wonder about or are curious about. These questions will be recorded on a
class chart or on the board and on the student recording sheet. Students will then use
mathematics to answer their own questions. Students will be given information to solve the
problem based on need. When they realize they don’t have the information they need, and ask
for it, it will be given to them.
Task Directions:
Act 1 – Whole Group - Pose the conflict and introduce students to the scenario by showing Act
I picture. (Dan Meyer http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/)
“Introduce the central conflict of your story/task clearly, visually, viscerally, using as few
words as possible.”
● Show Act 1 video- https://vimeo.com/94999740
● Share and record students’ questions. The teacher may need to guide students so that the
questions generated are math-related. Students may need to watch the video several
times.
Anticipated questions students may ask and wish to answer: (*Main question(s) to be
investigated)
● *How can the students split the cupcakes equally?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 38 of 100
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● What fraction of a cupcake will each person get?
● Once students have their question, ask the students to estimate answers to their questions
(think-pair-share). Students will write their best estimate, then write two more estimates –
one that is too low and one that is too high so that they establish a range in which the
solution should occur. Students should plot their three estimates on an empty number
line. Note: As the facilitator, you may choose to allow the students to answer their own
posed questions, one question that a fellow student posed, or a related question listed
above. For students to be completely engaged in the inquiry-based problem solving
process, it is important for them to experience ownership of the questions posed.
Important note: Although students will only investigate the main question(s) for this task, it
is important for the teacher to not ignore student generated questions. Additional questions
may be answered after they’ve found a solution to the main question, or as homework or
extra projects.
Act 2 – Student Exploration - Provide additional information as students work toward solutions
to their questions. (Dan Meyer http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/) “The protagonist/student overcomes obstacles, looks for resources, and develops new tools.”
● During Act 2, students decide on the facts, tools, and other information needed to answer
the question(s) (from Act 1). When students decide what they need to solve the problem,
they should ask for those things. It is pivotal to the problem-solving process that students
decide what is needed without being given the information up front.
● Required Information: (If students don’t get the information from the video)
o 4 students
o Only 3 cupcakes
● Some groups might need scaffolds to guide them. The teacher should question groups
who seem to be moving in the wrong direction or might not know where to begin.
Questioning is an effective strategy that can be used, with questions such as:
● What is the problem you are trying to solve?
● What do you think affects the situation?
● Can you explain what you’ve done so far?
● What strategies are you using?
● What assumptions are you making?
● What tools or models may help you?
● Why is that true?
● Does that make sense?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 39 of 100
All Rights Reserved
Act 3 – Whole Group – Share solutions and strategies.
● Students to present their solutions and strategies and compare them.
● Information for reveal:
o Each person will get ¾ of a cupcake
● Lead discussion to compare these, asking questions such as:
o How reasonable was your estimate?
o Which strategy was most efficient?
o Can you think of another method that might have worked?
o What might you do differently next time?
Act 4, The Sequel - “The goals of the sequel task are to a) challenge students who finished
quickly so b) I can help students who need my help. It can't feel like punishment for good work.
It can't seem like drudgery. It has to entice and activate the imagination.” Dan Meyer
http://blog.mrmeyer.com/2013/teaching-with-three-act-tasks-act-three-sequel/
For Act 4, share ideas below (see extensions) or reference other student-generated questions that
could be used for additional classwork, projects or homework.
FORMATIVE ASSESSMENT QUESTIONS
● What organizational strategies did you use?
● How did you know what fraction you had?
● Is it possible to show each fraction in a different way? Show me your fraction in a
different way.
DIFFERENTIATION
Extension
• Students can create their own “cupcake party problems” where they can experiment with
the number of people at the party and the number of cupcakes to be shared. Have
students write a journal entry explaining their findings. Where there any problems that
were easier to solve than others? Why?
Intervention
• Allow students to first determine how the four students could share 1 cupcake. Once the
student soles how to share 1 cupcake, allow them to expand on that knowledge to share the 3
cupcakes.
• Intervention Table
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 40 of 100
All Rights Reserved
Task Title: ________________________ Name: ________________________
Adapted from Andrew Stadel
ACT 1
What did/do you notice?
What questions come to your mind?
Main Question: ________________________________________________________________
What is your 1st estimate and why?
On an empty number line, record an estimate that is too low and an estimate that is too high.
ACT 2
What information would you like to know or need to solve the MAIN question?
Record the given information (measurements, materials, etc…)
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 41 of 100
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Act 2 (con’t)
If possible, give a better estimation with this information: _______________________________
Use this area for your work, tables, calculations, sketches, and final solution.
ACT 3
What was the result?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 42 of 100
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CONSTRUCTING TASK: USING FRACTION STRIPS TO EXPLORE THE
NUMBER LINE Return to Task Table
Adapted from a lesson by Michelle Clay, Floyd County, GA
APPROXIMATE TIME: 2 class periods
Students create fraction number lines using strips of paper and use the number lines to find
equalities and inequalities.
CONTENT STANDARDS
MGSE3.NF.2 Understand a fraction as a number on the number line; represent fractions on a
number line diagram.
a. Represent a fraction 1
𝑏 on a number line diagram by defining the interval from 0 to 1 as
the whole and partitioning it into b equal parts. Recognize that each part has size 1
𝑏.
Recognize that a unit fraction 1
𝑏 is located 1
𝑏 whole unit from 0 on the number line.
b. Represent a non-unit fraction 𝑎
𝑏 on a number line diagram by marking off a lengths of 1
𝑏
(unit fractions) from 0. Recognize that the resulting interval has size 𝑎
𝑏 and that its
endpoint locates the non-unit fraction 𝑎
𝑏 on the number line.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Children need to understand the meaning of fractions based on repeated hands-on activities.
They need a general rule for explaining the numerator and denominator of a fraction. They
need to understand that fractions are numbers that can be represented on a number line.
Students need to understand that fractions between 0 – 1 can have denominators and
numerators greater than one.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 43 of 100
All Rights Reserved
COMMON MISCONCEPTIONS
Students do not understand that when partitioning a whole shape, number line, or a set into unit
fractions, the intervals must be equal. Students also do not count correctly on the number line.
For example, students may count the hash mark at zero as the first unit fraction.
ESSENTIAL QUESTIONS
● What fractions are on the number line between 0 and 1?
● What relationships can I discover about fractions?
● How are tenths related to the whole?
MATERIALS
● Using Fraction Strips to Explore the Number Line Activity task sheet
● 9” x 12” sheets of paper in six different colors (cut into 1” x 12” strips) Each child will
need one strip of paper in each color.
● Scissors
● File folder (1 for each child) or math journal
● Glue or tape
GROUPING
Individual/Partner Task
NUMBER TALKS
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Students make and use a set of fraction strips to represent the interval between zero and one on
the number line, discover fraction relationships, and work with equivalent fractions.
Part I (SMP 4)
To begin the lesson, give students six strips of paper in six different colors. Specify one color
and have students hold up one strip of this color. Tell students that this strip will represent the
number line from zero to one. Have students glue or tape the strip to the back of their file folder
or math journal. The students will label folder above the left-hand side of the strip “0” and
above the right-hand edge of the strip “1.”
Next, ask students to pick a second strip and fold it into two equal pieces. Have students label
above this strip with the numerals 0, ½, 1.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 44 of 100
All Rights Reserved
Have students take out another strip, fold it twice, and divide it into four congruent pieces. Have
students label the space above the strip using 0, ¼, 2/4, ¾, 1. Repeat this process of folding,
cutting, and naming strips for thirds, and sixths. Have students use a ruler and label the last strip
in 12ths by drawing a line at every inch. This particular number line will represent 1 foot. The
inches are showing fractions of a foot. 1/12, 2/12, and so on.
Part II (SMP 1, 4, 5, 6, 7)
Arrange students in small groups of 2-3 students. Give them approximately ten minutes to write
down their observations from comparing the Number Lines. Have each group share some of
their comments. Lead the groups to consider questions such as:
● How are the Fraction Strips and Number Lines similar?
● How are they different?
Remind students that the fraction strip is equal to the length of a ruler which is one foot. Ask
students to label ½ a foot with the letter A. Ask students to label 2/3 of a foot with B. Continue
asking students to label fractional parts of a foot with letters.
Part III (SMP 1, 2, 3, 4, 5, 6, 7, 8)
Have students work in small groups to answer the questions below. The teacher should monitor
the groups, asking questions, and encouraging students to explore the concept of fractions on the
Number Line.
Have groups (at least 2-3) share their solution to question numbers 6 and 7. Try to pick groups
who presented different ways of solving the problems. After this lesson, have students store
their Fraction Strips in their sandwich bag.
FORMATIVE ASSESSMENT QUESTIONS
● What fractions are on the number line between 0-1?
● How did you determine the various fractions between 0-1?
DIFFERENTIATION
Extension
● Have students create additional strips representing fractions between 0 - 5 and write
about relationships.
Intervention
● Use ready-made Fraction Tiles or Virtual Manipulatives.
● Line ‘Em Up
Select four or five fractions for students to put in order from least to greatest. Have them
indicate approximately where each fraction belongs on the number line labeled only with
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 45 of 100
All Rights Reserved
the points 0 and 1. Adding machine paper can be used as a number line. Students can
compare their lines with others and explain how they decided where to place the
fractions.
Adapted from Elementary and Middle School Mathematics: Teaching Developmentally
By John A. Van de Walle, Karen S. Karp, and Jennifer M. Bay-Williams, p. 301.
• Intervention Table
TECHNOLOGY RESOURCES
• http://www.mathsisfun.com/numbers/fraction-number-line.html
• https://www.conceptuamath.com/app/tool-library Conceptua Learning Tools (Fraction
Tab) are great for both parents and teachers while working on fraction concepts
• http://www.visualfractions.com/FindGrampy/findgrampy.html
• https://www.brainpop.com/games/battleshipnumberline/
• http://www.sheppardsoftware.com/mathgames/fractions/AnimalRescueFractionsNumber
LineGame.htm
o Students estimate position on an empty number line in these engaging games.
• http://www.dreambox.com/k-8-math-lessons (scroll to “Placing Fractions on the
Number Line”) This lesson engages students in actively placing fractions in their correct
location on the number line. A number line representation ensures students understand
how to compare and order fractions apart from any specific part-whole context. Instead of
using a particular context, students use landmark fractions and numbers to place fractions
on a number line from 0 to 1 and from 0 to 2.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 46 of 100
All Rights Reserved
Name: _______________________________ Date: _______________
USING FRACTION STRIPS TO EXPLORE THE NUMBER LINE
(Adapted from a Learning Task by Michelle Clay, Floyd County, GA)
1. Using complete sentences and math words, write 3 observations you and your group
made about fractions between 0 and 1 on the Number Line.
Use your Number Lines to answer the following questions.
2. How many sixths are between 0 and 1?
3. How many 12ths are equivalent to 1 whole?
4. What fraction on the Number Line is equivalent to 2
6 ?
Put on your thinking caps….
5. If 3
3 is equivalent to the whole number 1, how many thirds are in the whole number 2?
6. What would the fraction 12
4 represent? Draw a picture in the space below to explain
your answer.
7. During a lesson on Measurement, students were asked to measure their feet using a
ruler. Lexi’s foot measured 7 inches. Addie’s foot was 5
6 of a foot. Robert’s foot was equal
to ¾ of a foot. Andrew’s foot measured 2
3 of a foot. Use your number line to help you
arrange the students’ foot measurements in order from smallest to largest. On the back
of this paper, sketch the Number Lines divided into thirds, fourths, sixths, and inches (1
12).
Use pictures, numbers, and words to explain your solution.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 47 of 100
All Rights Reserved
CONSTRUCTING TASK: I LIKE TO MOVE IT! MOVE IT!! Return to Task Table
APPROXIMATE TIME: 1-2 class periods
Students will count unit fraction on number lines.
CONTENT STANDARDS
MGSE3.NF.1 Understand a fraction 1𝑏 as the quantity formed by 1 part when a whole is
partitioned into b equal parts (unit fraction); understand a fraction ab as the quantity formed by a
parts of size 1/𝑏. For example, 3/4 means there are three 1/4 parts, so 3/4 = 1/4 + 1/4 + 1/4 . MGSE3.NF.2 Understand a fraction as a number on the number line; represent fractions on a
number line diagram.
MGSE3.OA.3 Use multiplication and division within 100 to solve word problems in situations
involving equal groups, arrays, and measurement quantities,‡ e.g., by using drawings and
equations with a symbol for the unknown number to represent the problem.12 ‡See Glossary:
Multiplication and Division Within 100.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Van de Walle stated that in whole-number learning, counting precedes and helps students
compare the size of numbers and later to add and subtract. This is also true with fractions.
Counting fractional parts, initially unit fractions, to see how multiple parts compare to the whole
helps students understand the relationship between the parts (the numerator) and the whole (the
denominator). (A unit fraction is a single fractional part. The fractions 1/3 and 1/8 are unit
fractions). Students should come to think of counting fractional parts in much the same way as
they; might count apples or other objects. If you know the kind of part you are counting
(denominator), you can tell when you get to one whole, when you get to two wholes, and so on.
Students should be able to answer the question, “How many fifths are in a whole?” us as they
know how many ones are in a ten. However, in the 2008 National Assessment of Education
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 48 of 100
All Rights Reserved
Progress (NAEP) only 44 percent of fourth grade students answered that question correctly
(Rampey, Dion, & Donahue, 2009).
The students will continue to be reinforced about their understanding of a unit fraction and the
similarities and differences between it and units of 1 (whole numbers). Thus, in this task, the
students will:
a. Develop an understanding of how, like units of 1, unit fractions are positioned on a
number line in patterns of odd and even.
b. Adding unit fractions together can make wholes with extra “parts” as remainders.
c. “Explore” b’s relationship to division with remainders.
Throughout this unit, students have justified their understanding of how unit fractions make a
whole. Just like counting whole numbers, the counting of unit fractions is called iterating. Like
partitioning, iterating is an important part of being able to understand and use fractions.
Understanding that ¾ can be thought of as a count of three parts called fourths is an important
idea for students to develop (Van de Walle).
COMMON MISCONCEPTIONS
Many students do not see unit fractions as the basic building block of fractions, in the same sense
that the number 1 is the basic building block of the whole numbers. Just as every whole number
is obtained by combining sufficient number of 1s, every fraction is obtained by combining a
sufficient number of unit fractions (Kentucky DOE FALs document).
ESSENTIAL QUESTIONS
● What is the relationship between a unit fraction and a unit of 1?
● How is the odd and even pattern with unit fractions on a number line similar to units of 1
on a number line?
● Why is the denominator important to the unit fractions?
● How does the numerator impact the denominator on the number line?
MATERIALS
● Sidewalk chalk
GROUPING
Partners/Small Group
NUMBER TALKS
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 49 of 100
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TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Part I
It is imperative that the teacher conducts the following discussion using the below mentioned
questions before the task since they will serve as a bridge for conceptual understanding.
On the board, the teacher will display the following number lines but have it continue to the
whole number 5 with ⅓ and ⅔ being labeled between each whole unit and likewise with sixths.
This will be done to reinforce how the denominator tells how many equal units it takes to make a
whole and size relationship.
Questions:
a. How many units are represented between 0 and 1 on the first number line? second
number line?
b. What’s the major difference between the two number lines?
c. What do you think would happen if I had 5 thirds? Could it be placed on the number
line?
d. Are there fractions that are at the same point on the number line?
e. Why do you think that happened?
The teacher can then ask questions and have the students position the fractions on the number
line. The Progressions document states that the words “proper” and “improper” fractions should
NOT be introduced initially; instead 5/3 is the “quantity” you get by combining 5 parts together
when the whole is divided into 3 equal parts. The Progressions third grade document also
states that the goal is for students to see unit fractions as the basic building blocks of fractions, in
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 50 of 100
All Rights Reserved
the same sense that the number 1 is the basic building block of the whole numbers; just as every
whole number is obtained by combining a sufficient number of 1s, every fraction is obtained by
combining a sufficient number of unit fractions.
The progressions’ document states that students need to see that unit fractions, the one with
the larger denominator is smaller, by reasoning, for example, that in order for more (identical)
pieces to make the same whole, the pieces must be smaller. From this they reason that for
fractions that have the same numerator, the fraction with the smaller denominator is greater.
For example, ⅖ >2/7, because 1/ 7 < ⅕, so 2 lengths of 1/7 is less than 2 lengths of ⅕ .
Part II (SMP 1, 2, 3, 4, 5, 6, 7, 8)
The teacher will then take the students outside to the parking lot or sidewalk. She will have the
students broken into groups of three. Each group will have been assigned a color which will
match the sidewalk chalk number lines on the ground. Each group will have four number lines
on the ground. The teacher will have this prepared with the number lines beginning at 0 and
extending to 5. Their first task will be to partition each number line into halves, thirds, fourths,
and sixths. Between each whole number should be 1/3, 2/3, or ¼, 2/4, ¾, etc. You never want
the student to lose sight of whatever the denominator is, that’s how many parts it takes to make a
whole. As for the number lines, be clear to say that each number line has its own fractions.
Halves are on a line by themselves, thirds are on their own line and sixths are on their own line.
They are NOT combined. Note for teacher: Creating unit fractions on a number line is
actually more challenging than it sounds. The students will be developing their conceptual
understanding of how halves should be drawn larger than thirds and so forth. Having a
student serve as a “resident expert” once the fractions lines are completed will also reinforce
the idea that the smaller the denominator the bigger the part and vice versa. The “expert”
could also have a sheet with a sample to serve as a visual if need be.
Also, the students could partition the wholes into units without writing the fraction underneath.
However, the students must know that between each whole number, the partitioning of units
MUST remain the same. 3rd grade progressions document, fractions, page 3
The teacher should position the students in the parking lot or sidewalk side by side but with
enough space so that each group can work independently. This is important too so that when
they begin the game, each team will have the same distance to run. The following gives an
example of the way it could look.
______No. Line Halves_______ _________No. Line Halves___________ _________No. Line Halves____ ______No. Line Thirds________ _________No. Line Thirds___________ _________No. Line Thirds____ ______No. Line Fourths_______ _________No. Line Fourths__________ _________No. Line Fourths____ ______No. Line Sixths________ _________No. Line Sixths___________ _________No. Line Sixths____
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 51 of 100
All Rights Reserved
Once the number lines are partitioned into their correct fractional parts, the teams then go and
stand at their starting line which should be about 8 to 10 feet directly behind their number lines
and side by side with the other teams for fairness. The teacher will then call out a fraction. The
first team member to run and stand on the correct position wins the point. They also must orally
count aloud 1/3, 2/3, 3/3, 4/3, 5/3 so that ALL students gain an understanding of how unit
fractions are counting numbers.
FORMATIVE ASSESSMENT
● What did you notice happening when the numerator was larger than the denominator?
● How is counting the number of parts related to creating whole numbers?
● How is that related to division?
DIFFERENTIATION
• Extension #1- Van de Walle Activity 12. 5, “How Far Did She Go?”, Teaching Student
Centered Mathematics, Second Edition
Give students number lines partitioned such that only some of the partitions are showing.
Meaning, between 0 and 1, give two partition fraction lines near zero, but leave the rest
off to see if they can determine that there were six equal groupings. The other number
line directly underneath could be eighths, but the partitioning lines could be near the
number one. The scenario could be use a context such as walking to school. For each
number line, ask, “How far has Nicole gone? How do you know? The teacher must have
one of the partitioning lines circled on each graph and require the student to justify their
responses to check for conceptual understanding. As an activity, the teacher can have the
students create their own number lines with just a few partitioning lines displayed either
near zero or one and have a classmate determine the location. Instead of it being just 0 to
1, it could be extended to whole numbers 5 or 10 depending on their readiness.
• Extension #2- This task could continue by doing similar activities from the previous day
but the students must tell you the location in wholes and part. For example, if they are
standing on 7/3 on the number line, then that’s 2 and 1/3. Note: This is NOT a third
grade standard. However, it is an excellent opportunity to “expose” and justify how the
parts of the denominator makes a whole and how the numerator factors in because the
numerator partners with the denominator to possibly make wholes with parts left over. In
fourth grade they call this a mixed number. This also shows a connection between
division and fractions due to the divisor and denominator sharing the same role and the
numerator is the dividend. This IS NOT meant for them to know, but to show
relationships on the progression.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 52 of 100
All Rights Reserved
Intervention
• The students pull out their fractional self-created sets from the previous lesson or teacher
provided fractional strips. The students will practice counting the fractional parts and
drawing out different amounts. For example, they could count and display ½, 2/2, 3/2,
4/2, and 5/2. They would then draw the representation to look like a number line and
label each part as typed above to show how you can count unit fractions like unit whole
numbers.
Just For Fun: The task is entitled “I like to move it! Move it!” When the students are using
the sidewalk chalk to partition each number line into unit fractions, play the Madagascar song
to get them moving since they will be running and moving throughout the game!
• Intervention Table
TECHNOLOGY RESOURCES
• https://www.conceptuamath.com/app/tool-library Conceptua Learning Tools (Fraction
Tab) are great for both parents and teachers while working on fraction concepts
• http://www.dreambox.com/k-8-math-lessons (scroll to “Placing Fractions on the
Number Line”) This lesson engages students in actively placing fractions in their correct
location on the number line. A number line representation ensures students understand
how to compare and order fractions apart from any specific part-whole context. Instead of
using a particular context, students use landmark fractions and numbers to place fractions
on a number line from 0 to 1 and from 0 to 2.
• http://www.visualfractions.com/FindGrampy/findgrampy.html
• https://www.brainpop.com/games/battleshipnumberline/
• http://www.sheppardsoftware.com/mathgames/fractions/AnimalRescueFractionsNumber
LineGame.htm
▪ Students estimate position on an empty number line in these engaging games.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 53 of 100
All Rights Reserved
CONSTRUCTING TASK: PATTERN BLOCK FRACTIONS REVISITED
Adapted from a Learning Task by Debra Childs, Floyd County, GA Return to Task Table
APPROXIMATE TIME: 1 class period
Students will partition pattern blocks using various sized wholes.
CONTENT STANDARDS
MGSE3.NF.1 Understand a fraction 1
𝑏 as the quantity formed by 1 part when a whole is
partitioned into b equal parts (unit fraction); understand a fraction 𝑎
𝑏 as the quantity formed by a
parts of size 1
𝑏. For example, 3
4 means there are three
1
4 parts, so 3
4 = 1
4 + 1
4 + 1
4 .
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
A big idea for students to explore is that fractional parts are equal-sized parts of a whole
unit. All the parts must be the same size. The unit can be a collection of things, and the unit is
counted as one. The names for fractions tell how many parts of that size are needed to make the
whole. In this activity, different wholes are designated in the same model. This discourages
children from identifying a fractional part with a special shape or color, challenging them to see
the relationship of each part to the designated whole. (Elementary and Middle School
Mathematics: Teaching Developmentally, John A. Van de Walle, Karen S. Karp, and Jennifer
M. Bay-Williams.)
COMMON MISCONCEPTIONS
Students do not understand that when partitioning a whole shape, number line, or a set into unit
fractions, the intervals must be equal. Another misconception is that students do not understand
the importance of the whole of a fraction and identifying it. For example, students may use a
fixed size of ¼ based on the manipulatives used or previous experience with a ruler. Students
also think all shapes can be divided the same way. These misconceptions will be addressed in
this task.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 54 of 100
All Rights Reserved
ESSENTIAL QUESTIONS
● How can I use pattern blocks to name fractions?
● How does the size of the whole affect the size of the fractions?
● Is ¼ always the same size? How do you know?
MATERIALS
● Pattern Blocks
● Exploring Fractions Further With Pattern Blocks Activity Sheet
GROUPING
Partner/Small Group Task
NUMBER TALKS
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Part I (SMP 4)
Begin this task by presenting the students with 2 different sized pieces of construction paper. (If
you wish, you can name it a pan of brownies or pizza.) Pose the question: Is ¼ always the same
size? With the students fold the larger piece of paper into fourths, then do the same with the
smaller piece. Verify with the students that each piece was evenly folded into fourths. Ask the
question, “Which ¼ of a pizza/pan of brownies would you like to have?” This will lead to a
discussion around the size of the whole.
Part II (SMP 1, 3, 4, 5, 7)
Lead students in a discussion including questions such as:
● What if you use two yellow hexagon blocks to represent the whole?
● What fractional part of the whole will one yellow hexagon be?
● What block will represent ¼? What other relationships do you see?
Have students work together to complete the task sheet. Students should model each question
with pattern blocks and make a sketch of the required blocks.
FORMATIVE ASSESSMENT QUESTIONS
● How did you determine ¼?
● How did the size of ½ change from the whole on page 1 to the whole on page 2?
● How does the size of the whole affect the size of the fractional piece?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 55 of 100
All Rights Reserved
● Can you find any equivalent fractions? How do you know?
DIFFERENTIATION
Extension
● Challenge students to explore additional variations of the whole such as three hexagons,
four trapezoids, or eight triangles.
Intervention
● Finding Fair Shares - Give students other objects or models and ask them to find halves
or fourths. Use familiar objects such as groups of cookies or pieces of candy. Vary the
number of pieces in the whole. (Teaching Student-Centered Mathematics, Grades 3-5,
John A. Van de Walle, Karen S. Karp, and Jennifer M. Bay-Williams, p. 136.
● Intervention Table
TECHNOLOGY RESOURCES
• https://www.conceptuamath.com/app/tool-library Conceptua Learning Tools (Fraction
Tab) are great for both parents and teachers while working on fraction concepts
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 56 of 100
All Rights Reserved
PATTERN BLOCK FRACTIONS Revisited:
EXPLORING FRACTIONS FURTHER WITH PATTERN BLOCKS
(Adapted from a Learning Task by Debra Childs, Floyd County, GA)
Name ____________________________________ Date _______________________
Task: Pattern Block Fractions
● Use the pattern blocks to solve
the riddles below.
● Draw the shape and label each fractional part.
If this is one whole, what is 1
2?
(Draw and label)
If this is one whole, what is 1
4?
(Draw and label)
If this is one whole, what is 1
6?
(Draw and label)
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 57 of 100
All Rights Reserved
If this is one whole, what is 1
2?
(Draw and label)
If this is one whole, what is 1
4?
(Draw and label)
If this is one whole, show 1 1
2.
(Draw and label)
Use pictures, words, and numbers to summarize what you have learned from this task.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 58 of 100
All Rights Reserved
3ACT TASK: PARTY TRAY Return to Task Table
APPROXIMATE TIME: 1 class period
CONTENT STANDARDS
MGSE3.NF.1 Understand a fraction 1
𝑏 as the quantity formed by 1 part when a whole is
partitioned into b equal parts (unit fraction); understand a fraction 𝑎
𝑏 as the quantity formed by a
parts of size 1
𝑏. For example, 3
4 means there are three
1
4 parts, so 3
4 = 1
4 + 1
4 + 1
4 .
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them. Students must make sense of
the problem by identifying what information they need to solve their question.
2. Reason abstractly and quantitatively. Students are asked to make an estimate (high
and low).
3. Construct viable arguments and critique the reasoning of others. After writing down
their own questions, students discuss their question with partners, creating the
opportunity to construct the argument of why they chose their question, as well as
critiquing the questions that others came up with.
4. Model with mathematics. Once given the information, the students use that
information to develop a mathematical model to solve their question. Students should
use a model to justify their reasoning.
5. Use appropriate tools strategically. Students use number lines, drawings, or
manipulatives to help them answer their main questions regarding fractions.
6. Attend to precision. Students use vocabulary such as numerator, denominator, and
fractions with increasing precision to discuss their reasoning.
8. Look for and express regularity in repeated reasoning. Students continually evaluate
their work by asking themselves “Does this make sense?”
ESSENTIAL QUESTIONS
In order to maintain a student-inquiry-based approach to this task, it may be beneficial to wait
until Act 2 to share the EQ’s with your students. By doing this, students will be allowed the
opportunity to be very creative with their thinking in Act 1. By sharing the EQ’s in Act 2, you
will be able to narrow the focus of inquiry so that the outcome results in student learning directly
related to the content standards aligned with this task.
● How are fractions used in problem-solving situations?
● How can I use fractions to name parts of a whole?
● What are the important features of a unit fraction?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 59 of 100
All Rights Reserved
MATERIALS
● Act 1 picture- Sandwich Tray
● Act 2 picture- Partitioned Sandwich
● Student recording sheet
GROUPING
Individual/Partner and or Small Group
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will view the picture and tell what they noticed. Next, they will be asked to
discuss what they wonder about or are curious about. These questions will be recorded on a
class chart or on the board and on the student recording sheet. Students will then use
mathematics to answer their own questions. Students will be given information to solve the
problem based on need. When they realize they don’t have the information they need, and ask
for it, it will be given to them.
BACKGROUND KNOWLEDGE:
This task follows the 3-Act Math Task format originally developed by Dan Meyer. More
information on this type of task may be found at http://blog.mrmeyer.com/category/3acts/. A
Three-Act Task is a whole-group mathematics task consisting of 3 distinct parts: an engaging
and perplexing Act One, an information and solution seeking Act Two, and a solution discussion
and solution revealing Act Three. More information along with guidelines for 3-Act Tasks may
be found in the Guide to Three-Act Tasks on georgiastandards.org and the K-5 Georgia
Mathematics Wiki.
Students are introduced to fractions in 1st and 2nd grade by partitioning circles and rectangles
into two, three, or four equal shares, describing the shares using the words halves, thirds, fourths,
and quarters.
Counting fractional parts is the groundwork for comparing and understanding the two parts of
fractions. When developing this thinking, it is useful to display fraction pie pieces and count
them together as a class. For example, using the fractions 1/4, 2/4, 3/4, 4/4, and 5/4, the class can
discuss the relationship the fractions have with one whole. (Van de Walle, p. 138) This task will
provide a foundation for exploring equivalent fractions in the next lessons of this unit. This task
also links fractions with the concept of division which they have learned about in an earlier 3rd
grade unit.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 60 of 100
All Rights Reserved
COMMON MISCONCEPTIONS:
Students do not understand the importance of the whole of a fraction and identifying it. For
example, students may use a fixed size of ¼ based on the manipulatives used or previous
experience with a ruler.
Task Directions:
Act 1 – Whole Group - Pose the conflict and introduce students to the scenario by showing Act
I picture. (Dan Meyer http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-story/)
“Introduce the central conflict of your story/task clearly, visually, viscerally, using as few
words as possible.”
● Show Act 1 picture – Sandwich Tray
● Share and record students’ questions. The teacher may need to guide students so
that the questions generated are math-related.
Anticipated questions students may ask and wish to answer: (*Main question(s) to be
investigated)
● What fraction of a sandwich is each piece?
● How many people can the tray feed?
● *How many whole sandwiches are on the tray?
● How many whole sandwiches are there of each type of sandwich?
● Once students have their question, ask the students to estimate answers to their questions
(think-pair-share). Students will write their best estimate, then write two more estimates –
one that is too low and one that is too high so that they establish a range in which the
solution should occur. Students should plot their three estimates on an empty number
line. Note: As the facilitator, you may choose to allow the students to answer their own
posed questions, one question that a fellow student posed, or a related question listed
above. For students to be completely engaged in the inquiry-based problem solving
process, it is important for them to experience ownership of the questions posed.
Important note: Although students will only investigate the main question(s) for this task, it
is important for the teacher to not ignore student generated questions. Additional questions
may be answered after they’ve found a solution to the main question, or as homework or
extra projects.
Act 2 – Student Exploration - Provide additional information as students work toward solutions
to their questions. (Dan Meyer http://blog.mrmeyer.com/2011/the-three-acts-of-a-mathematical-
story/)
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 61 of 100
All Rights Reserved
“The protagonist/student overcomes obstacles, looks for resources, and develops new
tools.”
● During Act 2, students decide on the facts, tools, and other information needed to answer
the question(s) (from Act 1). When students decide what they need to solve the problem,
they should ask for those things. It is pivotal to the problem-solving process that students
decide what is needed without being given the information up front.
● Required Information: Act 2 Picture- Partitioned Sandwich
● Some groups might need scaffolds to guide them. The teacher should question groups
who seem to be moving in the wrong direction or might not know where to begin.
Questioning is an effective strategy that can be used, with questions such as:
● What is the problem you are trying to solve?
● What do you think affects the situation?
● Can you explain what you’ve done so far?
● What strategies are you using?
● What assumptions are you making?
● What tools or models may help you?
● Why is that true?
● Does that make sense?
Act 3 – Whole Group – Share solutions and strategies.
● Students present their solutions and strategies and compare them.
● Information for reveal:
o 46 pieces that are ¼ in size
o Total 11 ½ or 11 2/4 sandwiches
● As students share their solutions, this is a great opportunity to have the discussion of
equivalent fractions. If a student gets the answer of 11 ½ sandwiches, but another gets 11
2/4 sandwiches, have students construct a viable argument for which student is correct or
are both students correct. Once the students establish that both students are correct have
the discussion of equivalence and what that means.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 62 of 100
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● Lead discussion to compare these, asking questions such as:
o How reasonable was your estimate?
o Which strategy was most efficient?
o Can you think of another method that might have worked?
o What might you do differently next time?
Act 4, The Sequel - “The goals of the sequel task are to a) challenge students who finished
quickly so b) I can help students who need my help. It can't feel like punishment for good work.
It can't seem like drudgery. It has to entice and activate the imagination.” Dan Meyer
http://blog.mrmeyer.com/2013/teaching-with-three-act-tasks-act-three-sequel/
For Act 4, share ideas below (see extensions) or reference other student-generated questions that
could be used for additional classwork, projects or homework.
FORMATIVE ASSESSMENT QUESTIONS
● What organizational strategies did you use?
● How is counting the number of parts related to creating whole numbers?
DIFFERENTIATION
Extension
● If each person ate 2/4 of a sandwich, how many people can the party tray feed?
● Pose similar problems to students. For example: If the tray contained 36 parts of a
sandwich and each part was 1/3 of a sandwich in size, how many sandwiches are on
the tray?
● If one person ate 1/3 of the sandwich pieces on the tray, how many pieces did they
eat? How many whole sandwiches did they eat?
Intervention
● Provide students with manipulatives to act out forming sandwiches from the parts on the
tray.
● Intervention Table
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 63 of 100
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Act 1 Picture- Sandwich Tray
Act 2 Picture- Partitioned Sandwich
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 64 of 100
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Task Title: ________________________ Name: ________________________
Adapted from Andrew Stadel
ACT 1
What did/do you notice?
What questions come to your mind?
Main Question: ________________________________________________________________
What is your 1st estimate and why?
On an empty number line, record an estimate that is too low and an estimate that is too high.
ACT 2
What information would you like to know or need to solve the MAIN question?
Record the given information (measurements, materials, etc…)
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 65 of 100
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Act 2 (con’t)
If possible, give a better estimation with this information: _______________________________
Use this area for your work, tables, calculations, sketches, and final solution.
ACT 3
What was the result?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 66 of 100
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PRACTICING TASK: MAKE A HEXAGON GAME Return to Task Table
Adapted from a Learning Task from K-5 Math Teaching Resources
APPROXIMATE TIME: 1 class period
Students will play a game where they create a fraction with dice and build their
fraction on hexagons using pattern blocks.
CONTENT STANDARDS
MGSE3.NF.1 Understand a fraction 1
𝑏 as the quantity formed by 1 part when a whole is
partitioned into b equal parts (unit fraction); understand a fraction 𝑎
𝑏 as the quantity formed by a
parts of size 1
𝑏. For example, 3
4 means there are three
1
4 parts, so 3
4 = 1
4 + 1
4 + 1
4 .
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
The way we write fractions with a top and a bottom number with a bar between is simply an
arbitrary agreement (convention) for how to represent fractions. It falls into the category of
things you simply tell/show students. However, students do need to know the meaning of the
numerator and denominator. The numerator tells how many shares or parts we have, how many
have been counted, how many we are talking about. It counts the parts or shares. The
denominator tells what is being counted. It tells what fractional part is being counted such as
fourths or sixths.
(Elementary and Middle School Mathematics: Teaching Developmentally, John A. Van de
Walle, Karen S. Karp, and Jennifer M. Bay-Williams.)
COMMON MISCONCEPTIONS
Students see the numbers in fractions as two unrelated whole numbers separated by a line.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 67 of 100
All Rights Reserved
ESSENTIAL QUESTIONS
● How can I use pattern blocks to name fractions? MATERIALS
● Pattern Blocks: hexagon, triangles, trapezoid, blue rhombi
● Build a Hexagon Instructions and Game Board for each player
● Dice
GROUPING
Partner Task
NUMBER TALKS
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION (SMP 1, 2, 3, 4, and 5)
Instruct students to work with a partner. For the game, students will take turns rolling two dice.
The largest number rolled is the denominator and the smaller number is the numerator. Students
build the fractional amount rolled on the game board using pattern blocks. Students may use
equivalent fractions. If students roll a fraction they cannot build, they lose a turn. Play continues
until one player has covered all the hexagons on his game board.
FORMATIVE ASSESSMENT QUESTIONS
● What does the top number (numerator) tell us?
● What does the bottom number (denominator) tell us?
● What happened in the game if you rolled the same number on both dice?
● Did you have to trade triangles for other shape blocks? What equal trades did you make?
DIFFERENTIATION
Extension
● Modify the game by changing the whole. Try using two hexagons for the whole or three
trapezoids for the whole.
Intervention
● More, Less, or Equal to One Whole – Give students a collection of fractional parts (all
the same type) and indicate the kind of fractional part they have. Parts can be drawn on a
worksheet or physical models placed in plastic baggies with an identifying card. For
example, if done with fraction strips, the collection might have seven strips with a card
indicating “these are eighths.” The task is to decide if the collection is less than one
whole, equal to a whole, or more than a whole. Students must draw pictures and/or use
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 68 of 100
All Rights Reserved
numbers to explain their answer. They can also tell how close the set is to a complete
whole. (Teaching Student-Centered Mathematics, Grades 3-5, John A. Van de Walle,
Karen S. Karp, and Jennifer M. Bay-Williams, p. 138.)
● Intervention Table
TECHNOLOGY RESOURCES
• https://www.conceptuamath.com/app/tool-library Conceptua Learning Tools (Fraction
Tab) are great for both parents and teachers while working on fraction concepts
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 69 of 100
All Rights Reserved
Build a Hexagon
Materials: game board for each player, dice, pattern blocks
(hexagon, triangles, trapezoids, blue rhombi)
Work with a partner. Take turns to roll two dice. The largest number you roll is the
denominator and the smaller number is the numerator.
1. Use pattern blocks to build the fractional amount you rolled on the game board.
You may use equivalent fractions.
2. If you roll a denominator that you can’t build, you lose a turn.
3. Keep going until one player has covered all the hexagons on his/her game board.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 70 of 100
All Rights Reserved
Build a Hexagon
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 71 of 100
All Rights Reserved
CONSTRUCTING TASK: PIZZAS MADE TO ORDER Return to Task Table
Adapted from a Learning Task by Cara Coker, Floyd County, GA
APPROXIMATE TIME: 1-2 class periods
Students will fill pizza orders by representing the ordered ingredients on the
appropriate fractional parts of a pizza cut-out.
CONTENT STANDARDS
MGSE3.NF.1 Understand a fraction 1
𝑏 as the quantity formed by 1 part when a
whole is partitioned into b equal parts (unit fraction); understand a fraction 𝑎
𝑏 as the quantity
formed by a parts of size 1
𝑏. For example, 3
4 means there are three
1
4 parts, so 3
4 = 1
4 + 1
4 + 1
4 .
MGSE3.NF.3 Explain equivalence of fractions through reasoning with visual fraction models.
Compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point
on a number line.
b. Recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6, and
8, e.g., 1
2 =
2
4,
4
6 =
2
3. Explain why the fractions are equivalent, e.g., by using a visual
fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole
numbers. Examples: Express 3 in the form 3 = 6
2 (3 wholes is equal to six halves);
recognize that 3
1 = 3; locate
4
4 and 1 at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning
about their size. Recognize that comparisons are valid only when the two fractions refer
to the same whole. Record the results of comparisons with the symbols >, =, or <, and
justify the conclusions, e.g., by using a visual fraction model.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 72 of 100
All Rights Reserved
BACKGROUND KNOWLEDGE
Before the activity, be sure the children understand the concept of equal parts. Practice with the
student’s methods to divide various shapes into fractional pieces. Have students practice
drawing lines to divide squares, rectangles, triangles, and circles into halves, fourths, eighths.
COMMON MISCONCEPTIONS
Students do not understand there are many fractions less than 1. Students do not understand
fractions can be greater than 1.
ESSENTIAL QUESTIONS
● How can I represent fractions of different sizes?
● What relationships can I discover about fractions?
● What is a real-life example of using fractions?
MATERIALS
● Give Me Half! By Stuart J. Murphy (or another book about the concept
of fractions).
● Scissors
● Glue or paste
● Crayons
● One large sheet of black paper
● One half sheet of brown paper
● Small pieces of various colored paper including red, white, green, yellow, black
● Pizza Order Directions – One per child
GROUPING
Individual Task
NUMBER TALKS
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Part I
To assess prior knowledge, brainstorm with students about food that is divided into equal pieces.
Possible suggestions may include a chocolate bar, apple pie, pizza, and an orange. Read aloud
and discuss, Give Me Half! By Stuart J. Murphy (or another book about the concept of fractions).
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 73 of 100
All Rights Reserved
Part II (SMP 1, 2, 4, 5)
To begin the lesson, give students a half sheet of brown paper. Instruct them to draw and cut out
a circle from the brown paper. Then give each child a Pizza Order. Instruct the students to use
their pencil to divide their circles into the fractional part used in the Pizza Order (fourths or
eights). Then have the students trace over their pencil lines with a dark crayon. Next, give
students small sheets of the colored paper (red, white, green, yellow, black). Instruct students to
cut pieces of the colored paper to represent the pizza toppings. The toppings should be glued
onto the appropriate number of pizza slices.
After the toppings have been successfully glued to the brown circle, give each student a sheet of
black construction paper. Have the students glue their pizzas and Pizza Order Directions to the
paper.
FORMATIVE ASSESSMENT QUESTIONS
● What fraction of your pizza is covered with peppers?
● What topping covers most of your pizza?
● Are black olives covering more or less than half your pizza?
● How did you divide your pizza into equal parts?
● How many equal parts did you need? How did you know?
● If your whole pizza was divided into fourths, how many slices did you cover with
toppings? How would you write this as equivalent fractions? (4/4 = 1)
● If your pizza is covered with 1/8 mushrooms and 3/8 green peppers, does it have more
mushrooms or green peppers? How do you know? (Encourage students to explain in
terms of the pizza size and by comparing numerators in the fraction.)
● Some of you covered 4/8 of your pizzas with pepperoni. Can you name equivalent
fractions for 4/8?
● Were any pizzas covered with ½ cheese? Why did your Pizza Order ask for 2/4 cheese?
● Do you see any other examples of equivalent fractions on the pizzas?
DIFFERENTIATION
Extension
• Have students create additional pizzas using more challenging fractional parts such
as thirds, sixths, tenths. Increase the number of toppings. Have some sections
contain more than one topping.
Intervention
• Provide ready-cut circles and if necessary, draw dotted lines for students to trace as
they divide their pizzas into fractional parts. Have students complete Pizza Orders
using fractions containing only common denominators.
• Intervention Table
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 74 of 100
All Rights Reserved
TECHNOLOGY RESOURCES
http://mrnussbaum.com/pizza_game/index.html
http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/fractions/index.htm
http://www.primarygames.com/fractions/2a.htm
Student Work Sample
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 75 of 100
All Rights Reserved
PIZZAS MADE TO ORDER: PIZZA ORDER
DIRECTIONS
Adapted from a lesson by Cara Coker, Floyd County, GA
----------------------------------------------------------
I would like to order a pizza that is 1/8 green peppers,
8/8 pepperoni, and 3/8 mushrooms.
----------------------------------------------------------
I would like to order a pizza that is
¼ mushrooms, 2/4 cheese, and ¼ pepperoni.
----------------------------------------------------------
I would like to order a pizza that is
1/8 black olives, 8/8 mushrooms, and
4/8 pepperoni.
----------------------------------------------------------
I would like to order a pizza that is
¼ mushrooms, ¼ black olives, and
½ pepperoni.
----------------------------------------------------------
I would like to order a pizza that is
¼ cheese, ¼ black olives, ¼ pepperoni, and
¼ green peppers.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 76 of 100
All Rights Reserved
CONSTRUCTING TASK: GRAPHING FRACTIONS Return to Task Table
From NCTM Illuminations
APPROXIMATE TIME: 2 class periods
Students will draw a picture graph of their classmates’ favorite pets, a bar
graph of their classmates’ favorite sports, and a graph of their choice of a
bag of colored candies. Students will identify the fractional representation of each bar of data
and create questions that could be answered using the data in their graphs.
CONTENT STANDARDS
MGSE3.MD.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with
several categories. Solve one- and two-step “how many more” and “how many less” problems
using information presented in scaled bar graphs. For example, draw a bar graph in which each
square in the bar graph might represent 5 pets.
MGSE3.NF.1 Understand a fraction 1
𝑏 as the quantity formed by 1 part when a whole is
partitioned into b equal parts (unit fraction); understand a fraction 𝑎
𝑏 as the quantity formed by a
parts of size 1
𝑏. For example, 3
4 means there are three
1
4 parts, so 3
4 = 1
4 + 1
4 + 1
4 .
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Children need to understand the meaning of fractions based on repeated hands-on activities.
They need a general rule for explaining the numerator and denominator of a fraction.
Students should be familiar with various types of graphs including bar graphs, and line plots.
Students may not realize that data can be described and displayed using fractions.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 77 of 100
All Rights Reserved
COMMON MISCONCEPTIONS
Students may read the mark on a scale that is below a designated number on the scale as if it was
the next number. For example, a mark that is one mark below 80 grams may be read as 81 grams.
Students realize it is one away from 80, but do not think of it as 79 grams.
Although intervals on a bar graph are not in single units, students count each square as one. To
avoid this error, have students include tick marks between each interval. Students should begin
each scale with 0. They should think of skip-counting when determining the value of a bar since
the scale is not in single units.
ESSENTIAL QUESTIONS
● How can I collect and organize data?
● How can I display fractional parts of data in a graph?
MATERIALS
Small individual bag of candy for each student
GROUPING
Individual/Partner Task
NUMBER TALKS
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
Part I (SMP 1, 2, 4, 5, 7)
As a class or in small groups, create a picture graph of favorite pets. An example is shown
below.
FAVORITE PETS
Cat
Dog
Hamster
Each stands for 2 votes.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 78 of 100
All Rights Reserved
Have students determine the fractional representation for pet. For example, in the graph shown
above there are:
● 10 children out of 20 prefer cats (10/20),
● 4 children out of 20 prefer dogs (4/20),
● 6 children out of 20 prefer hamsters (6/20)
Discuss the graph. As a class, create problems that could be answered using the data. You may
want to display the graph for other classes to analyze.
Part II (SMP 1, 2, 3, 4, 5, ,6, 7)
As a class, create a bar graph of students’ favorite sports. An example is shown below,
Once again, have students determine the fractional representation for favorite sports. For
example, in the graph shown above there are:
● 9 children out of 22 prefer soccer (9/22)
● 4 children out of 22 prefer softball (4/22)
● 6 children out of 22 prefer basketball (6/22)
● 3 children out of 22 prefer other sports (3/22)
Discuss the graph, and create questions that can be answered using the data.
Part III (SMP 1, 2, 4, 5, 6, 7)
Working in small groups, students will examine the set model of fractions using colored candies.
Give students an individual bag or pack of colored candies. Have students open their bag of
candies and sort by color. Have students count the number of each color in their group and
record the data in table on notebook paper. Have students record the fraction of each color
represented in their group.
Have students log on to the Create a Graph Tool from the National Center for Education
Statistics. Students should choose the type of graph they want to create by using the pull-down
menu. Once students have created and printed their graph, they should label the data in fractional
parts.
Have students work in groups to create story problems relating to their graphs. Examples of
problems students might write:
● Which group had the most candies?
● How many candies did they have in their pack?
● What is the difference between the greatest and least number of candies in a pack?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 79 of 100
All Rights Reserved
FORMATIVE ASSESSMENT QUESTIONS
● Which type of graph did you create when you went to the Create a Graph Tool from the
National Center for Educational Statistics?
● Why did you select this type of graph?
DIFFERENTIATION
Extension
• Have students create more than one graphical representational of the candy data.
Discuss which display is most effective in presenting the data.
Intervention
• Have students graph fewer pieces of candy using sticky notes to represent elements
of data in a student-created graph.
Adapted from Elementary and Middle School Mathematics: Teaching Developmentally
By John A. Van de Walle, Karen S. Karp, and Jennifer M. Bay-Williams, p. 443.
• Intervention Table
TECHNOLOGY RESOURCES
• https://www.conceptuamath.com/app/tool-library Conceptua Learning Tools (Fraction
Tab) are great for both parents and teachers while working on fraction concepts
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 80 of 100
All Rights Reserved
CONSTRUCTING TASK: INCH BY INCH Return to Task Table
APPROXIMATE TIME: Two class periods
Students will measure using strips of paper (non-standard units). Students will build a
ruler beginning with inch units that they will glue to tagboard. Students will practice measuring
with their ruler and compare it to a standard ruler. Students will create a line plot graph to collect
and record the data of the objects they have measured to the nearest ¼ inch throughout the class.
CONTENT STANDARDS
MGSE3.MD.4 Generate measurement data by measuring lengths using rulers marked with
halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is
marked off in appropriate units – whole numbers, halves, or quarters.
(Refer to grade level overview for unpacked standards)
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
If students actually make simple measuring instruments using unit models with which
they are familiar, it is more likely that they will understand how an instrument measures. A ruler
is the most important measurement tool that primary students need to learn about. If students
line up physical units, such as paper clips, along with a strip of tag board, and mark them off,
they can see that it is the spaces on rulers, not the marks or numbers that are important. It is
essential that the measurement with actual unit models be compared with measurement with
using an instrument. The temptation is to carefully explain to students how to use these units to
measure and then send them off to practice measuring. This approach will shift students’
attention to the procedure (following your instruction) and away from developing an
understanding of measurement using units. (Van de Walle, p.72-72)
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 81 of 100
All Rights Reserved
COMMON MISCONCEPTIONS
Students plot points based on understanding fractions as whole numbers instead of fractional
parts. For example: Students order fractions using the numerator or students order unit fractions
by the denominator.
ESSENTIAL QUESTIONS
● What estimation strategies are used in measurement?
● How is the appropriate unit for measurement determined?
● How is the reasonableness of a measurement determined?
● Why are units important in measurement?
● How can I determine length to the nearest ¼ or ½?
MATERIALS
● tag board
● paper
● ruler
GROUPING
Whole group or small group
NUMBER TALKS
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
Part I (SMP 4, 5, 6, 7)
Cut strips of paper lengthwise (1-inch wide) from regular paper. Ask students what is a half and
how could they find where a half is on the strip. Students will fold the strip in half. Have the
students mark ½ on the folded line. Ask students: if we needed to make this strip into 4 equal
pieces/parts how could we do that? Allow for exploration. Students will see that by folding a ½
in ½ it makes ¼.
Discussion: allow students to see that the strip is folded into 4 parts. Explain that the first fold is
where 1 out of the four parts ends, have students label the next one 2/4 (two of 4 parts). Allow
the conversation to take place that the second line is already labeled ½ and now it is going to be
labeled 2 of 4 (2/4). What does that mean? Students will identify that they’re the same. Label
the last fold 3 of 4 or ¾.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 82 of 100
All Rights Reserved
Have students measure objects all around the room to the nearest ¼ strip and record their
findings in their journal. The strip is a non-standard unit of measurement and should be
recognized as such. Have the students measure things that are longer than one strip to count a
whole strip plus part. Example: the width of the desk is 2 strips and ¼ of a strip long.
Part II (SMP 1, 2, 4, 5, 6, 7)
Give the students a 1x1 inch square and repeat the entire process from day one. Allow students
time to measure using a single inch square. Students will recognize how tedious it is to measure
with a single inch square, and in many cases inaccurate. Give students (12) 1x1 squares and
mark them into fourths with a pencil (not folded).
Place the folded inch square on top of the blank inch squares as a template for marking and place
a dash to identify ¼ , 2/4, (1/2), and 3/4. Keep asking students what ½ and 2/4 have in common.
Give students a tag board strip that is 1 inch wide. Have students create an inch ruler by gluing
the MARKED 1x1 inch squares side by side.
After each student has created the ruler, have them measure things around the class using their
newly created inch ruler. After 10-15 minutes of exploring engage students in a discussion in
regards to the difficulties they encountered using their ruler (not labeled correctly, always had to
count what square it was, etc)
Introduce a ruler with inches. Discuss and compare the similarities between the created ruler and
the actual 12’ ruler. Discuss how the actual ruler is more accurate and efficient. Have the student
circulate the class measuring objects to the nearest ¼ of an inch using the actual ruler. Students
create a line plot graph to collect and record the data of the objects they have measured
throughout the class.
FORMATIVE ASSESSMENT QUESTIONS
● How did you determine ¼, ½, and ¾ on your strip of paper?
● Why is it important to have a standard unit of measurement?
● How does your “ruler” compare to the standard 12’ ruler?
● Looking at your line plot graph, which measurement seems to be the most
common among the classroom?
DIFFERENTIATION
Extension
● Measure around the classroom to the nearest ¼, ½ and whole inch using a broken
ruler.
Intervention
● Spend additional time with the original strip from part one.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 83 of 100
All Rights Reserved
● Have students create an additional strip that is a different size and determine the
¼, ½, and ¾ marks. Have students compare the two strips and lead a discussion
of the importance of standard measuring tools.
● Intervention Table
TECHNOLOGY RESOURCES
• https://www.conceptuamath.com/app/tool-library Conceptua Learning Tools (Fraction
Tab) are great for both parents and teachers while working on fraction concepts
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 84 of 100
All Rights Reserved
CONSTRUCTING TASK: MEASURING TO THE HALF AND QUARTER INCH
Adapted from a Learning Task from K-5 Math Teaching Resources Return to Task Table
APPROXIMATE TIME: 4 Class Periods
Students will measure objects to the nearest ½ and ¼ inch. Students will order their
measurements from shortest to longest and will create a line-plot graph to represent their
data.
CONTENT STANDARDS
MGSE3.MD.4 Generate measurement data by measuring lengths using rulers marked with
halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is
marked off in appropriate units— whole numbers, halves, or quarters.
MGSE3.NF.2 Understand a fraction as a number on the number line; represent fractions on a
number line diagram.
b. Represent a non-unit fraction 𝑎
𝑏 on a number line diagram by marking off a lengths
of 1
𝑏 (unit fractions) from 0. Recognize that the resulting interval has size 𝑎
𝑏 and that its
endpoint locates the non-unit fraction 𝑎
𝑏 on the number line.
MGSE3.NF.3 Explain equivalence of fractions through reasoning with visual fraction models.
Compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same
point on a number line.
b. Recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6,
and 8, e.g., 1
2 =
2
4,
4
6 =
2
3. Explain why the fractions are equivalent, e.g., by using a
visual fraction model.
d. Compare two fractions with the same numerator or the same denominator by reasoning
about their size. Recognize that comparisons are valid only when the two fractions refer
to the same whole. Record the results of comparisons with the symbols >, =, or <, and
justify the conclusions, e.g., by using a visual fraction model.
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 85 of 100
All Rights Reserved
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Fractions and measurement can be very difficult concepts for children to understand. This task
helps to combine fraction and measurement skills in a concrete and tangible activity geared
toward learners of every type. By using fractions while measuring objects, children will be able
to reason that fractions express a relationship between a part and a whole. Encourage students to
consider the fact that a ruler is simply a number line used as a measuring tool. They will begin
to see and apply fractions in their everyday lives as well as other areas of mathematics.
COMMON MISCONCEPTIONS
Students do not count correctly on the number line. For example, students may count the hash
mark at zero as the first unit fraction. This can hinder their ability to read a ruler.
ESSENTIAL QUESTIONS
● How can I organize data measured to the half inch?
● How can I organize data measured to the quarter inch?
MATERIALS
● Half-Inch and Quarter-Inch Ruler Templates
● Unlined Paper
GROUPING
Individual/Partner Task
NUMBER TALKS
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
Part 1 (SMP 1, 2, 4, 5, and 6)
To begin the lesson, give each child a half-inch ruler template. Instruct students to label their rulers to
show all half-inch measurements. Working individually or in small groups, have students use the rulers
to measure ten objects in the classroom to the nearest half-inch.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 86 of 100
All Rights Reserved
On a sheet of paper, students should sketch and label each object they measured. Then ask students to
number the objects in order from shortest to longest. Discuss how students decided which objects were
smaller and which objects were larger. Encourage discussion about comparing the physical size of each
object as well as comparing the fractional measurements of each object.
Part II (SMP 5)
Students use their data to create a line plot where the horizontal scale is marked off whole
numbers and halves. Completed line plots should be shared with small groups or entire class.
Classroom Objects Measured to Nearest Half-Inch
X
X X X
X X X X X X
______________________________________________________________________________
0 ½ 1 ½ 2 2 ½ 3 3 ½ 4 4 ½ 5 5 ½ 6
Part III (SMP 1, 2, 4, 5, 6)
Repeat the activity using quarter-inch measurement. Give each child a quarter-inch ruler template.
Instruct students to label their rulers to show all quarter-inch measurements. Ask students to write both
measures (ex. 2/4 and ½) for equivalent fractions on the ruler. Include discussion about equivalent
fractions in measurement. Working individually or in small groups, have students use the rulers to
measure ten objects in the classroom to the nearest quarter-inch.
On a sheet of paper, students should sketch and label each object they measured. Then ask students to
number the objects in order from shortest to longest. Discuss how students decided which objects were
smaller and which objects were larger. Encourage discussion about comparing the physical size of each
object as well as comparing the fractional measurements of each object.
Students use their data to create a line plot where the horizontal scale is marked off with whole
numbers, halves, and quarters. Completed line plots should be shared with small groups or entire
class.
FORMATIVE ASSESSMENT QUESTIONS
● In what ways is your ruler similar to a number line?
● How did you label your number line to the half inch and quarter inch?
● Which measure is more exact?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 87 of 100
All Rights Reserved
DIFERENTIATION
Extension
● Measure around the classroom to the nearest ¼, ½, and whole inch using a broken ruler.
Intervention
● Spend additional time with the original strip from part one.
● Have students create an additional strip that is a different size and determine the ¼, ½,
and ¾ marks. Have students compare the two strips and lead a discussion of the
importance of standard measuring tools.
● Intervention Table
TECHNOLOGY RESOURCES
• https://www.conceptuamath.com/app/tool-library Conceptua Learning Tools (Fraction
Tab) are great for both parents and teachers while working on fraction concepts
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 88 of 100
All Rights Reserved
Half-Inch and Quarter-Inch Ruler Templates
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 89 of 100
All Rights Reserved
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 90 of 100
All Rights Reserved
PRACTICE TASK: TRASH CAN BASKETBALL Return to Task Table
Adapted from a 1st Grade GPS Frameworks Task
APPROXIMATE TIME: 1 Class Period
Students will play a game where they record a tally mark each time they shoot a
trash ball into a trashcan. Students will write a fraction that represents the
number of shots made and then create a poster that represents their results using
an inequality.
CONTENT STANDARDS
MGSE3.NF.1 Understand a fraction 1
𝑏 as the quantity formed by 1 part when a whole is
partitioned into b equal parts (unit fraction); understand a fraction 𝑎
𝑏 as the quantity formed by a
parts of size 1
𝑏. For example, 3
4 means there are three
1
4 parts, so 3
4 = 1
4 + 1
4 + 1
4 .
MGSE3.NF.2 Understand a fraction as a number on the number line; represent fractions on a
number line diagram.
a. Represent a fraction 1
𝑏 on a number line diagram by defining the interval from 0 to 1 as
the whole and partitioning it into b equal parts. Recognize that each part has size 1
𝑏.
Recognize that a unit fraction 1
𝑏 is located 1
𝑏 whole unit from 0 on the number line.
b. Represent a non-unit fraction 𝑎
𝑏 on a number line diagram by marking off a lengths of 1
𝑏
(unit fractions) from 0. Recognize that the resulting interval has size 𝑎
𝑏 and that its
endpoint locates the non-unit fraction 𝑎
𝑏 on the number line.
MGSE3.NF.3 Explain equivalence of fractions through reasoning with visual fraction models.
Compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point
on a number line.
b. Recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6, and 8,
e.g., 1
2 =
2
4,
4
6 =
2
3. Explain why the fractions are equivalent, e.g., by using a visual fraction
model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole
numbers. Examples: Express 3 in the form 3 = 6
2 (3 wholes is equal to six halves);
recognize that 3
1 = 3; locate 4
4 and 1 at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning
about their size. Recognize that comparisons are valid only when the two fractions refer to
the same whole. Record the results of comparisons with the symbols >, =, or <, and justify
the conclusions, e.g., by using a visual fraction model.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 91 of 100
All Rights Reserved
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
Students have learned to write fractions as part of a whole and part of a group. They have also
learned to compare fractions. This task allows students to practice their new knowledge in a
game format.
COMMON MISCONCEPTIONS
Students do not understand the importance of the whole of a fraction and identifying it. For
example, students may use a fixed size of ¼ based on the manipulatives used or previous
experience with a ruler.
ESSENTIAL QUESTIONS
● How can I write a fraction to represent a part of a group?
● When we compare two fractions, how do we know which has a greater value?
MATERIALS
● “Trash Can Basketball” student recording sheet
● Each group will need 10 pieces of “trash” (paper balls).
● Box, tub, or trash can for a container
● Crayons or markers and construction paper for making a poster
GROUPING
Partner/Small Group Activity
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 92 of 100
All Rights Reserved
NUMBER TALKS
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION (SMP 1, 2, 4, 5, and 6)
Students collect data from playing “Trash Can Basketball.” They use the data to write and
compare fractions.
1. Students use scrap paper to make 10 paper balls per group. (Wad the paper balls up
tightly so they are easier to aim.)
2. Place a trash can (or other large container) 5 feet away.
3. Students predict how many paper balls they will be able to get into the basket. Predictions
should be written in the chart on the student recording sheet.
4. Students take turns with their partner(s) throwing the ten paper balls into the trash can.
The partner will collect data using tally marks on the chart to show how many of the 10
paper balls went into the trash can.
The copy room is a good source of trash paper. Be sure the paper balls are tight. Loosely
packed ones make it really difficult to throw accurately. Before beginning the throwing
contest, as a class, decide on any rules regarding practice throws.
FORMATIVE ASSESSMENT QUESTIONS
● How did you determine your score? How many times did you throw the paper ball?
How many times did you “make a basket”?
● How did you compare your fraction to your opponent’s?
DIFFERENTIATION
Extension
● Repeat the activity as time permits. (Try different types of paper balls,
distances, types of shots, etc.)
Intervention
● Have the chart pre-made on the poster for student use and/or allow student to
write his/her results on a computer, print, and attach to the poster.
TECHNOLOGY RESOURCES
• http://www.mathsisfun.com/numbers/fractions-match-words-pizza.html
• https://www.conceptuamath.com/app/tool-library Conceptua Learning Tools (Fraction
Tab) are great for both parents and teachers while working on fraction concepts
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 93 of 100
All Rights Reserved
Name ______________________________ Date ___________________________
TRASH CAN BASKETBALL
This is your chance to demonstrate your basketball skills! You have been
chosen to participate in a paper-ball throwing contest.
Directions:
1. Use the scrap paper to make 10 paper balls per group. (Wad the paper
balls up tightly so they are easier to aim.)
2. Place a trash can (or other large container) 5 feet away.
3. Predict how many paper balls you will be able to get into the basket. Write your prediction
in the chart below.
4. Take turns with your partner(s) throwing the ten paper balls into the trash can. Your
partner will collect data using tally marks on the chart below to show how many of the 10
paper balls went into the trash can.
Player #1
_______________
Number of
Tosses
Prediction for
Number of
“Baskets”
Number of
“Baskets”
(Use tallies)
Fraction of Baskets
Made
10
Player #2
_______________
Number of
Tosses
Prediction for
Number of
“Baskets”
Number of
“Baskets”
(Use tallies)
Fraction of Baskets
Made
10
5. On a sheet of unlined paper, create a poster to display your group’s results. Your poster
should include the following. Write to explain the results of the contest. Be prepared to
share your poster and results with the class. Represent the number of good throws for each
partner as a fraction and express a comparison of fraction scores using a >, <, or = symbol.
Make your poster colorful and informative!
Example:
Player #1 6
10
Player # 2 7
10
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 94 of 100
All Rights Reserved
6
10<
7
10
PERFORMANCE TASK: THE FRACTION STORY GAME Return to Task Table
APPROXIMATE TIME: 1-2 class periods
Students create a game while reviewing all the different aspects of fractions
they have studied.
CONTENT STANDARDS
MGSE3.NF.1 Understand a fraction 1
𝑏 as the quantity formed by 1 part when a whole is
partitioned into b equal parts (unit fraction); understand a fraction 𝑎
𝑏 as the quantity formed by a
parts of size 1
𝑏. For example, 3
4 means there are three
1
4 parts, so 3
4 = 1
4 + 1
4 + 1
4 .
MGSE3.NF.2 Understand a fraction as a number on the number line; represent fractions on a
number line diagram.
a. Represent a fraction 1
𝑏 on a number line diagram by defining the interval from 0 to 1 as
the whole and partitioning it into b equal parts. Recognize that each part has size 1
𝑏.
Recognize that a unit fraction 1
𝑏 is located 1
𝑏 whole unit from 0 on the number line.
b. Represent a non-unit fraction 𝑎
𝑏 on a number line diagram by marking off a lengths of 1
𝑏
(unit fractions) from 0. Recognize that the resulting interval has size 𝑎
𝑏 and that its
endpoint locates the non-unit fraction 𝑎
𝑏 on the number line.
MGSE3.NF.3 Explain equivalence of fractions through reasoning with visual fraction models.
Compare fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point
on a number line.
b. Recognize and generate simple equivalent fractions with denominators of 2, 3, 4, 6, and
8, e.g., 1
2 =
2
4,
4
6 =
2
3. Explain why the fractions are equivalent, e.g., by using a visual
fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole
numbers. Examples: Express 3 in the form 3 = 6
2 (3 wholes is equal to six halves);
recognize that 3
1 = 3; locate 4
4 and 1 at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning
about their size. Recognize that comparisons are valid only when the two fractions refer
to the same whole. Record the results of comparisons with the symbols >, =, or <, and
justify the conclusions, e.g., by using a visual fraction model.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 95 of 100
All Rights Reserved
STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
BACKGROUND KNOWLEDGE
While this task may serve as a summative assessment, it also may be used for teaching and
learning. It is important that all elements of the task be addressed throughout the unit so that
students understand what is expected of them.
COMMON MISCONCEPTIONS
Students do not understand that when partitioning a whole shape, number line, or a set into unit
fractions, the intervals must be equal.
ESSENTIAL QUESTION
● How are fractions used in problem-solving situations?
MATERIALS
• Materials Required Per Group
● “The Fraction Story Game, Directions” student sheet
● “The Fraction Story Game, Game board” student sheet
● Colored pencils or crayons
● Index cards (about 60)
● Common classroom materials -
Recycled items for game pieces
(about 6)
GROUPING
Small Group Task
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 96 of 100
All Rights Reserved
NUMBER TALKS
By now number talks should be incorporated into the daily math routine. Continue utilizing the
different strategies in number talks and revisiting them based on the needs of your students.
TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION
Students create a game while reviewing all the different aspects of fractions they have
studied.
Comments
Students may not understand what you mean by “common classroom materials.” While
many classrooms have standard dice that can be used, give alternative examples such as a
penny can be flipped to determine how many spaces the players get to move (heads = 2
spaces, tails =1 space). For game pieces, extra marker caps, plastic soda lids, manipulatives,
or coins can be used.
Part I (SMP 1, 2, and 6)
Begin by having students review lessons or activities that have been done during the fraction
unit. Record their thoughts on chart paper or the board. You may want to post a list of the
elements of the standards covered during the unit and reflect on tasks and activities which
addressed each element.
Students will write 20 – 30 word problems that assess the standards covered. You may want
the children to work with a partner or in small groups to create enough questions.
This culminating task represents the level of depth, rigor, and complexity expected of all third
grade students to demonstrate evidence of learning.
Additional Comments:
● Students should have had multiple opportunities to write story problems by this
time in the school year.
● Questions should match a standard.
● Index cards may be used for the problem cards. Insist that the students write
legibly. All problem cards should have the solutions on the back
● Solutions should be accompanied by an explanation/illustration.
● Game boards, playing pieces, and cards can be stored in large Ziploc bags or
manila folders.
The cards students create for their games can be used in a variety of ways. The problem
cards can be used to create a Jeopardy type game which can be played as a review of the
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 97 of 100
All Rights Reserved
unit. Also, the problem cards can be reproduced and used as a set of review question before
the unit assessment.
Part II (SMP 1, 2, 3, 4, 5, 6, 7, and 8)
Students will follow the directions below from “The Fraction Story Game, Directions” student
sheet.
Your task is to create a fraction story game using what you learned about fractions. Use the
fraction game board on “The Fraction Story Game, Game Board” student sheet to create a game
that other students will want to play.
Directions:
● Look at the list of the standard that you studied in class. The problem cards
you create must match those standards.
● You will need to make approximately 30 problem cards for your game. Most
of the cards should be written in story problem form.
● Be sure you have some problem cards for each of the standards addressed in
this unit.
● Each problem card must have the correct answer on the back. Cover each
problem card with a blank index card so players cannot see the problems
before their turn. See sample below.
● Write the rules for your game.
Things to
remember:
• You can only use common classroom materials.
● You may decorate your game board in a way that makes the game interesting
and fun to play.
● Be sure to play your game with a partner to be sure it works.
FORMATIVE ASSESSMENT QUESTIONS
● What are the skills you learned during this unit?
● What kind of problem can you create for ____ (one of the elements of the standard)?
● How do you know this is the correct solution for your problem?
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 98 of 100
All Rights Reserved
DIFFERENTIATION
Extension
● Students can create their own game board format with penalties, rewards, and more
complex rules.
Intervention
● Allow students to work in a small group so each student will need to make only
one card per standard.
● For some of the parts of a standard, give the students the problem and require
them to create the solution to the problem.
● Intervention Table
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 99 of 100
All Rights Reserved
Name ______________________________________ Date ___________
THE FRACTION STORY GAME
Your task is to create a fraction story game using what you learned about common
fractions and decimal fractions. Use the fraction game board on “The Fraction Story
Game, Game Board” student sheet to create a game that other students will want to play.
Directions:
● Look at the list of the standards that you studied in class. The problem cards you
create must match the standard.
● You will need to make approximately 30 problem cards for your game. Most of the
cards should be written in story problem form.
● Be sure you have some problem cards for each of the standards addressed in this
unit. Make sure you use both fractions in your problem cards.
● Each problem card must have the correct answer on the back. Cover each problem
card with a blank index card so players cannot see the problems before their turn.
See sample below.
● Write the rules for your game.
Things to remember:
● You can only use common classroom materials.
● You may decorate your game board in a way that makes the game interesting and
fun to play.
● Be sure to play your game with a partner to be sure it works.
Georgia Department of Education
Georgia Standards of Excellence Framework GSE Representing and Comparing Fractions • Unit 5
Mathematics GSE Third Grade Unit 5: Representing and Comparing Fractions
Richard Woods, State School Superintendent
July 2017 Page 100 of 100
All Rights Reserved
Name ______________________________________ Date __________________
The Fraction Story Game
Game Board
Start
Finish
Problem Cards